Note:
My math journal.
Last updated: 2021-11-23
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The first case is a strong inversion, the second is a 2-periodic involution (given by twisting about a core torus).
One can assume that \(\tau\) is rotation about some axis.
There is an extension of \(\tau\) to \({\mathbb{B}}^4\), so define an equivariant slice surface \(\Sigma\) if \(\tau \Sigma = \Sigma\), and define an equivariant (slice?) genus as the minimal genus among such surfaces \(\tilde g_4(K)\)
Study \(\tilde g_4(K) - g_4(K)\). Boyle-Issan show this difference is unbounded for a family of periodic knots.
Prove a similar theorem: given \((K, \tau)\), define a set of numerical invariants using Floer homology which are
Produced a family of strongly invertible slice knots where \(\tilde g_4\) is unbounded.
Most (small crossing) knots admit a strong inversion.
Next: how to apply this machinery to seemingly non-equivariant things.
A slice surface \(\Sigma\) is isotopy equivariant iff \(\tau_{{\mathbb{B}}^4} \isotopic \Sigma\) rel boundary. Define isotopy equivariant genus \(\tilde{ig}_4(K)\) as the minimal genus of such \(\Sigma\).
Calculating this invariants gives a way of finding non-isotopic surfaces for \(K\).
Recent work: topologically isotopic but not smoothly isotopic surfaces.
Proving topologically isotopic: a known theorem involving equivalence of \(\pi_1\).
Theorem: produced a knot where \(\tilde{ig}_4(K) > 0\).
Does \({\mathbb{B}}^4\) actually matter here? The answer is no, can take \(\operatorname{ZHB}^4\).
A generalized isotopy equivariant surface is a triple \((W, \tau_W, \Sigma)\) where
Another application: let \(\Sigma, \Sigma'\) be two slices surfaces in \({\mathbb{B}}^4\) for \(K\). Interpolate: take \(\Sigma = \Sigma_0 \to \Sigma_1 \to \cdots \to \Sigma_n = \Sigma'\) where each arrow is a stabilization or destabilization or isotopy rel \(K\). How many arrows are needed? Define this as \(M_{{~\mathrel{\Big\vert}~}}(\Sigma, \Sigma')\), the stabilization number.
Theorem: for any \(m\), produce a knot \(J_m\) with two slice disks with stabilization distance exactly \(m\).
Theorem: if \((K, \tau)\) is strongly invertible slice and \(\Sigma\) is any slice disk for \(K\), then \(M_{{~\mathrel{\Big\vert}~}}(\Sigma, \tau_{{\mathbb{B}}^4} \Sigma) \geq \cdots\), some function of the numerical invariants.
These all induce maps on \(\CFK(K)\), where the \(\tau\) action induces a \(\tau\) action on \(\CFK(K)\). Isotopy equivariant knot cobordisms \(K_1\to K_2\) induce \(\tau{\hbox{-}}\)equivariant maps \(\CFK(K_1) \to \CFK(K_2)\) in the sense that this commutes with the two different \(\tau\) actions on either side.
Can use this to find knots that are concordant but not equivariantly concordant by using algebraic restrictions on bigraded \(\CFK(K)\)
Doing this with higher order diffeomorphisms: the roadblock is defining \(\operatorname{HF}\) mod \(p\)!
A nice modern intro to homotopy theory: https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/intro-homoto.pdf
Quotients are colimits:
Geometric realization as a \begin{align*}\[coend\end{align*} ]
Homotopy fibers:
Homotopy cofiber:
Spectra as a presentable \begin{align*}\[infty-category\end{align*} ]
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Hector Pasten, UGA NT seminar.
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Torelli: the map sending a curve to its Jacobian is an injection on points.
Intermediate Jacobian: introduce to prove irrationality of cubic threefolds. An abelian variety the parameterizes degree zero cycles in dimension 1, up to rational equivalence.
Relationship between complex projective and geometry and symplectic topology: Kähler manifolds.
Abouzaid: interesting results about symplectic topology of Hamiltonian fibrations over the 2-sphere, and their consequences for smooth projective maps over the projective line.
The Grothendieck group of mixed Hodge modules, which enhances the Grothendieck group of \(G{\hbox{-}}\)modules.
A motivic semiorthogonal decomposition is the decomposition of the derived category of a quotient stack \begin{align*}X/G\end{align*} into components related to the “fixed-point data.” They represent a categorical analog of the Atiyah-Bott localization formula in equivariant cohomology, and their existence is conjectured for finite G
Can define curvature and 2nd fundamental form for algebraic varieties?
Invariants like HOMFLY: invariants of quantum matrices
consider the stack of representations, its inertia stack and the nilpotent version of the inertia stack.
Hurwitz spaces H_{k,g}, parametrizing degree k, genus g covers of P^1
Kobayashi–Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite–Einstein metric if and only if the bundle is slope polystable
predicted that given two vector bundles V_1, V_2 whose first Chern classes both vanish and whose second Chern classes agree, the resulting line bundles Thom(V_1) and Thom(V_2) should agree in Pic(Ell_G(X)).
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Hannah Turner, GT: Branched Cyclic Covers and L-Spaces
Two main constructions for 3-manifolds: Dehn surgery and branched cyclic covers
Idea: \(C_n\curvearrowright M\), take quotient to get an \(n{\hbox{-}}\)fold covering map away from a branch locus (usually a knot or link).
Given a knot \(K\hookrightarrow S^3\), can produce a canonical cyclic branched cover for any \(n\), \(\Sigma_n(K)\).
Dehn surgeries: classified by \(p/q \in {\mathbb{Q}}\).
Fact: \(\dim_{{\mathbb{F}}_2} \widehat{\operatorname{HF}}(M) \geq \# H_1(M; {\mathbb{Z}})\) unless it’s infinite, in which case we set the RHS to zero. We say \(M\) is an \(L{\hbox{-}}\)space if this is an equality.
Conjecture: non \(L{\hbox{-}}\)space if and only if admits a co-oriented taut foliation (decomposition into surfaces) iff \(\pi_1\) is left orderable.
Q: push through local system correspondence, what does this say about reps \(\pi_1\to G\)..? Or local systems..?
We know foliation \(\implies\) non \(L{\hbox{-}}\)space, the other directions are all wide open.
Diagrams for knots: boxes with numbers are half-twists, sign prescribes directions.
Which branched covers of knots are \(L{\hbox{-}}\)spaces?
Nice trick: quotient by a \(C_2\) action to make it a double branched cover \(X\to X/C_2\), and find an \(n{\hbox{-}}\)fold branched cover \(\tilde X\to X\). Then take an \(n{\hbox{-}}\)fold branched cover \(\tilde{X/C_2} \to X/C_2\) and then its 2-fold branched cover will be \(\tilde X \to \tilde{X/C_2}\).
Weakly quasi-alternating \(K\): \(\Sigma_2(K)\) is an \(L{\hbox{-}}\)space.
There are tools for showing the \(\pi_1\) you get here are not left-orderable. Showing left-orderability: fewer tools, need representation theory.
Generalized \(L{\hbox{-}}\)space: \(L{\hbox{-}}\)spaces for \(S^1\times S^2\).
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Differential forms on (derived) \begin{align*}\[stacks\end{align*} ]:
What is non-commutative geometry?
Category of singularities:
Bloch’s conductor conjecture:
On harmonic bundles:
Existence of spin and string structures: kind of like applying a functor to the Whitehead tower and asking for sections of the image tower:
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UGA AG Seminar: Eloise Hamilton?
GIT: \(G\) a reductive group (trivial unipotent radical), \(G\curvearrowright X\) a projective variety, a lift of the action to an ample line bundle \({\mathcal{L}}\to X\) so that \(G\) “acts on functions on \(X\).”
Making GIT work more generally in non-reductive settings: adding a \({\mathbb{G}}_m\) grading seems to fix most issues!
Definition: \(H = U\rtimes R\in {\mathsf{Alg}}{\mathsf{Grp}}\) linear with \(U\) unipotent and \(R\) reductive is internally graded if there is a 1-parameter subgroup \(\lambda: k^{\times}\to Z(R)\) such that the adjoint action of \(\lambda(k^{\times})\curvearrowright\operatorname{Lie}U\) (the Lie algebra) has strictly positive weights.
Of interest: the \begin{align*}\[hyperbolicity conjecture\end{align*} ]. Call a projective variety over \({\mathbb{C}}\) Brody hyperbolic if any entire holomorphic map \({\mathbb{C}}\to X\) is constant.
\(\widehat{U}\) theorem can be used in situations addressed by classical GIT, e.g. curves, vector bundles or sheaves, Higgs bundles, quiver reps, etc. There is a notion of semistability in classical situations, and this allows defining moduli for unstable things. Really gives a moduli space parameterizing “stable” objects of a fixed instability type. Gives a stratification by instability types.
Prismatic cohomology: a \(p{\hbox{-}}\)adic analog of crystalline cohomology
Carries a Frobenius action.
\(H^i_{\prism}(\mathfrak{X}_{/ { \mathfrak{S} }} )\) is finitely generated over \(\mathfrak{S} = W { \left[\left[ {u} \right] \right] }\), some Witt ring?
\(\phi_{\prism}\) is a semilinear operator.
Any torsion must be \(p{\hbox{-}}\)power torsion, i.e. \(H^i_{\prism}({\mathfrak{X}}_{/ { {\mathfrak{S}}}} )_{{\operatorname{tors}}} = H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[p^{\infty}]\).
The pathological bits in all integral \(p{\hbox{-}}\)adic Hodge theories come from \(H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[u^{\infty}]\).
To study finite flat \(p{\hbox{-}}\)power group schemes, study their Dieudonne modules
Idk I just like this:
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Lev Tovstopyat-Nelip, “Floer Homology and Quasipositive Surfaces,” MSU.
\begin{align*}\[contact structure\end{align*} ] on an oriented \begin{align*}\[3-manifold\end{align*} ] \(Y\): a maximally nonintegrable 2-place field \(\xi\) where \(\xi = \ker( \alpha)\) for some \(\alpha\in \Omega^1(Y)\) with \(\alpha \vee d\alpha > 0\).
A transverse knot is a knot positively transverse to \({ \left.{{ \alpha }} \right|_{{K}} } > 0\).
A knot \(K \subseteq (Y, \xi)\) is \begin{align*}\[Legendrian\end{align*} ] if \({\mathbf{T}}K\leq { \left.{{\xi}} \right|_{{K}} }\), so \({ \left.{{\alpha}} \right|_{{K}} } = 0\).
A disk \({\mathbb{D}}^2 \subseteq (Y, \xi)\) is \begin{align*}\[overtwisted\end{align*} ] if \({{\partial}}{\mathbb{D}}^2\) is Legendrian, i.e. \({ \left.{{{\mathbf{T}}{\mathbb{D}}^2}} \right|_{{{{\partial}}{\mathbb{D}}^2}} } = { \left.{{\xi }} \right|_{{{\mathbb{D}}^2}} }\).
Eliashberg: overtwisted contact structures can be studied using algebraic topology (every homotopy class of plane fields contains an overtwisted contact structures)
Tight contact structures are the interesting ones!
Define self-linking number \(\sl(\widehat{B}) = w(B) - n\).
A type of \begin{align*}\[adjunction inequality\end{align*} ] - If \(K\) is transverse in \((S^3, \xi_{\text{std}})\) then \(\sl(K) \leq 2g(K) - 1\).
For \(\Sigma\) an oriented surface with connected \(\phi\in {\operatorname{MCG}}(\Sigma, {{\partial}}\Sigma)\), define \(Y_{\phi} \mathrel{\vcenter{:}}= S\times I / (x,1) \sim (\phi(x), 0)\).
Yields \((\Sigma, \phi)\) an open book decomposition.
There is a correspondence between open book decompositions on \(Y\) and contact structures on \(Y\).
Let \(\Sigma \hookrightarrow Y\) be a Seifert surface, then it is quasipositive with respect to \(\xi\) if there exists an o.b.d. \((S, \phi)\) such that \(\Sigma \subseteq S\) is \(\pi_1{\hbox{-}}\)injective.
Lyon: every Seifert surface in a closed oriented 3-manifold is quasipositive with respect to some contact structure \(\xi\).
Measure how far a knot is from being \begin{align*}\[fibred\end{align*} ]: fibre depth.
If \(K\) is semi-quasipositive with respect to \(\xi_\text{std}\), then \(\mkern 1.5mu\overline{\mkern-1.5mu\sl\mkern-1.5mu}\mkern 1.5mu(K) = 2g(K) - 1\) Interesting #open_question: does the converge hold? I.e. if \(\mkern 1.5mu\overline{\mkern-1.5mu\sl\mkern-1.5mu}\mkern 1.5mu(K) = 2g(K) - 1\), is \(K\) semi-quasipositive?
\(\widehat{\operatorname{HFK}}\) detects genus in the sense that \(g(K)\) is the maximal nonvanishing \(\widehat{\operatorname{HFK}}(-S^3, K, i)\).
See fiberedness detection and sutured knot Floer homology.
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https://arxiv.org/pdf/1904.06756.pdf
Some notes on \begin{align*}\[quadratic differentials\end{align*} ]:
Moduli space of \begin{align*}\[abelian differentials\end{align*} ] on a curve may be isomorphic to the moduli space f stability structures on the Fukaya category of the curve.
These moduli spaces admit good “wall and chamber” decompositions, with \begin{align*}\[wall crossing\end{align*} ] formulas due to Kontsevich.
Important theorems: vanishing of cohomology for \begin{align*}\[line bundles\end{align*} ] and existence of meromorphic sections:
A \begin{align*}\[principal divisor\end{align*} ] is a divisor of a meromorphic function. Taking \(\operatorname{Div}(X) / \mathop{\mathrm{Prin}}\operatorname{Div}(X)\) yields \({ \operatorname{Cl}} (X)\) the \begin{align*}\[divisor class group\end{align*} ] of \(X\).
There is a map \(\operatorname{Div}: {\operatorname{Pic}}(X) \to { \operatorname{Cl}} (X)\) sending a line bundle to its divisor class. This is an iso!
A meromorphic function has the same number of zeros and poles, i.e. \(\deg D = 0\) for \(D\in \mathop{\mathrm{Prin}}\operatorname{Div}(X)\), so degrees are well-defined for \({ \operatorname{Cl}} (X)\).
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Kristin DeVleming, UGA AG seminar talk on moduli of quartic \begin{align*}\[K3 surfaces\end{align*} ].
Jiuya Wang’s, UGA NT seminar talk
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Refs: \begin{align*}\[Advice\end{align*} ]
See Manin’s universal \begin{align*}\[quantum groups\end{align*} ].
Manin defines a universal bialgebra for \(A\), which coacts on \(A\) in a universal way.
See \begin{align*}\[koszul dual\end{align*} ]
Forgetful functor from Hopf algebras to bialgebras has a left adjoint: the \begin{align*}\[Hopf envelope\end{align*} ].
Universal quantum group: take \begin{align*}\[Hopf envelope\end{align*} ] of universal bialgebra.
See \begin{align*}\[quadratic algebra\end{align*} ]
Twisting conditions for bialgebras: \(B\) is \({\mathbb{Z}}{\hbox{-}}\)graded and \(\Delta(B_n) \subseteq B\tensorpower{2}\).
Zhang twist: supplies a twisted multiplication.
Possibly related to \begin{align*}\[alpha twisted vector space\end{align*} ]?
\(\grmod{A} { { \, \xrightarrow{\sim}\, }}\grmod{A^{\phi}}\) for \(A^{\phi}\) a Zhang twist.
Morita-Takeuchi equivalence: equivalence of categories of comodules.
This talk compares cocycle twists to Zhang twists.
For \({\mathcal{O}}(G)\) the coordinate ring of \(G\in{\mathsf{Alg}}{\mathsf{Grp}}\), elements \(g\in G\) induce automorphism \(r_g, \ell_g: {\mathcal{O}}(G){\circlearrowleft}\) by left/right translation, and every twisting pair is of the form \((r_g, \ell_g^{-1})\).
Sovereign: equivalence between left and right duality functors.
Pointed algebra: simple comodules are 1-dimensional
Smash product of Hopf algebras: \(H_1\otimes H_2\) as a vector space, with a deformed multiplication.
See \begin{align*}\[quantum Yang Baxter\end{align*} ] equations. Solutions are \(R\in \mathop{\mathrm{End}}_k(V{ {}^{ \scriptstyle\otimes_{2}^{)} } }\) satisfying a tensor formula corresponding to moving strands in a braid.
Can be obtained from any braiding on \({{H}{\hbox{-}}\mathsf{coMod}}\).
Use equivalence of braided monoidal cats to get new solutions: \(\cmods{H} { { \, \xrightarrow{\sim}\, }}{{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]}{\hbox{-}}\mathsf{coMod}}{ { \, \xrightarrow{\sim}\, }}{{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]^{\sigma}}{\hbox{-}}\mathsf{coMod}}\).
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A lead on how to formulate de Rham with a non-flat vector bundle c/o Arun: https://arxiv.org/abs/2006.02922
Hopfological algebra is related to \(d^p = 0\)?
\begin{align*}\[mock modular forms\end{align*} ], \begin{align*}\[moonshine\end{align*} ], \begin{align*}\[elliptic cohomology\end{align*} ]
Tags: #knots #concordance #geometric_topology
\(K_1, K_2\) are smoothly concordant iff there exists a smoothly embedded cylinder \(S^1\times I \hookrightarrow S^3\times I\) with \({{\partial}}(S^1\times I) = K_1 {\textstyle\coprod}-K_2\). The concordance group \(C\) is the abelian group given by knots \(K \hookrightarrow S^3\) under connect sum, modulo concordance.
If \(K_i \hookrightarrow Y_i \in \mathbb{Z}\operatorname{HS}^3\), then the \(K_i\) are homologically concordant if there is smoothly embedded cylinder \(S^1\times I \hookrightarrow W\) with \({{\partial}}(W, S^1\times I) = (Y_1, K_1) {\textstyle\coprod}(Y_2, K_2)\) with \(W\) a homology cobordism:
This yields a homological concordance group \(\widehat{C}_{\mathbb{Z}}\).
There is an injection (?) \(C_{\mathbb{Z}}\hookrightarrow\widehat{C}_{\mathbb{Z}}\) which is known by Levine not to be surjective. What can be said about the cokernel?
See \begin{align*}\[Seifert fibered space\end{align*} ], \begin{align*}\[ZHS3\end{align*} ]. These are all \begin{align*}\[homology cobordant\end{align*} ] to \(S^3\).
Proof uses \begin{align*}\[CFK\end{align*} ], a \({\mathbb{F}}[u, v]{\hbox{-}}\)module.
A knot-like complex over \(R\) is a complex \(C \in {\mathsf{gr}\,}_{{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }} \mathsf{Ch}(R)\) such that
Some examples: the knot Floer complex \begin{align*}\[CFK\end{align*} ] over a knot, \(\CFK_{{\mathbb{F}}[u, v]}(K)\). Theorem: every such complex is locally equivalent to a unique standard complex. Concordant knots produce locally equivalent complexes \(\CFK_R(K)\) for \(R \mathrel{\vcenter{:}}={\mathbb{F}}[u] \otimes_{\mathbb{F}}{\mathbb{F}}[z] / \left\langle{uv}\right\rangle\).
Set \(\mathsf{C} \mathrel{\vcenter{:}}={\operatorname{Emb}}(S^1, S^3)\), add the monoidal structure \({\sharp}\) for connect sum. Take “isotopy” category instead of homotopy category? The unit is \(\one = U\), the unknot up to isotopy. What is the stabilization of \({-}{\sharp}X\) for fixed choices of \(X\)? Or of other interesting functors? #idle_thoughts
See \begin{align*}\[torus knot\end{align*} ]
Can do base changes \(\CFK_{{\mathbb{F}}[u, v]}(M_n, K_n) \otimes_{{\mathbb{F}}[u, v]} \mathcal{X} \leadsto \CFK_{\mathcal{X}}(M_n, K_n)\) (may also need to change basis to get standard complex).
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\begin{align*}\[monad\|comonads\end{align*} ] in \(\mathsf{C}\): \begin{align*}\[coalgebra object\end{align*} ] in \([\mathsf{C}, \mathsf{C}]\).
\begin{align*}\[Comodules\end{align*} ] over a comonad \(T\): an object \(X\), a map \(a^\sharp: X\to TX\), and some coherence conditions. Often called \(T{\hbox{-}}\)algebras, called the category of \(T{\hbox{-}}\)comodules \({{T}{\hbox{-}}\mathsf{coMod}}(\mathsf{C})\).
A fun but non-obvious consequence of https://stacks.math.columbia.edu/tag/06WS: for \(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}\) faithfully flat, there is an equivalence of categories \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G) { { \, \xrightarrow{\sim}\, }}{\mathsf{Rep}}(G) ,\end{align*} the category of regular \(G{\hbox{-}}\)representations, i.e. \({{\Gamma(G)}{\hbox{-}}\mathsf{coMod}}\). See \begin{align*}\[regular representation\end{align*} ].
Why this is true:
\begin{align*}\[Barr-Beck\end{align*} ] says \(\tilde L\) is an equivalence under suitable conditions (\(L, R\) \begin{align*}\[conservative\end{align*} ] with \(L\) preserving \begin{align*}\[equalizers\end{align*} ]).
Set up the \begin{align*}\[adjunction\end{align*} ] \begin{align*} \mathsf{D} \mathrel{\vcenter{:}}=\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \mathrel{\vcenter{:}}=\mathsf{C} .\end{align*} Then \(LR \mathrel{\vcenter{:}}= p^*p_*\), and Barr-Beck yields \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G)\underset{\tilde{p^*}}{{ { \, \xrightarrow{\sim}\, }}} {{(p^*p_*)}{\hbox{-}}\mathsf{coMod}}({\mathsf{QCoh}}(\operatorname{Spec}R)) .\end{align*}
Use that if \(G\in{\mathsf{Aff}}{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}\) then \(\Gamma(G) \in \mathsf{Hopf}{\mathsf{Alg}_{/R}}\). Set \(\mathsf{C} \mathrel{\vcenter{:}}={\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), and \(F\in [\mathsf{C}, \mathsf{C}]\) to be \(F({-}) \mathrel{\vcenter{:}}=({-})\otimes_R \Gamma(G)\). Then there is an \begin{align*}\[equivalence of categories\end{align*} ] \begin{align*} {{F}{\hbox{-}}\mathsf{coMod}}(\mathsf{C}) { { \, \xrightarrow{\sim}\, }}{\mathsf{Rep}}(G) .\end{align*}
Then show that \(F\) is equivalent to \(p^*p_*\).
http://individual.utoronto.ca/groechenig/stacks.pdf #references
Refs: \begin{align*}\[stack\|stacks\end{align*} ] \begin{align*}\[vector bundles\|vector bundle\end{align*} ] \begin{align*}\[descent data\end{align*} ]
There is an equivalence of categories \({\mathsf{{\mathbb{R}}}{\hbox{-}}\mathsf{Mod}} { { \, \xrightarrow{\sim}\, }}\mathsf{Tw}{\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) where the latter consists of objects which are pairs \((V, f:V\to V)\) where \(f(\lambda \mathbf{v}) = \mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu \mathbf{v}\) is a structure map and \(f^2 = \operatorname{id}_V\) and morphisms \(\phi:V\to W\) that commute with the structure maps.
For field extensions \(L_{/ {k}}\), the ring morphism \(k\hookrightarrow L\) yields \(\operatorname{Spec}L \to \operatorname{Spec}k\), which behaves like a \begin{align*}\[covering space\end{align*} ] with \(\mathop{\mathrm{Deck}}(\operatorname{Spec}L _{/ {\operatorname{Spec}k}} ) \cong { \mathsf{Gal}} (L_{/ {k}} )\).
Vector bundles on \(\operatorname{Spec}k\) correspond to \({\mathsf{k}{\hbox{-}}\mathsf{Mod}}\), and Galois-equivariant vector bundles on \(\operatorname{Spec}L\) will correspond to vector bundles on the quotient \(\operatorname{Spec}k\).
\(R\in {\mathsf{Alg}}_{/ {A}}\): a ring morphism \(A\to R\).
Given \(f\in {\mathbb{Z}}[x_1,\cdots, x_n]\), taking the zero locus in a ring \(R\) yields a functor \(\mathsf{CRing}\to {\mathsf{Set}}\). To do this with \(f\in A[x_1,\cdots, x_n]\) for \(A\in \mathsf{CRing}\), one needs \(R\in {\mathsf{Alg}}_{/ {A}}\), so this yields a functor \({\mathsf{Alg}}_{/ {A}} \to {\mathsf{Set}}\).
Think of spaces as functors \(X\in [\mathsf{CRing}, {\mathsf{Set}}]\), then \(\operatorname{Spec}R \mathrel{\vcenter{:}}=\mathsf{CRing}(R, {-})\), so \(R\) corepresents \(\operatorname{Spec}R\) in \(\mathsf{CRing}\).
Can represent \(R \left[ { \scriptstyle { {f}^{-1}} } \right] = R[t]/\left\langle{tf-1}\right\rangle\).
Standard open subfunctors: \(\operatorname{Spec}R \left[ { \scriptstyle { {f_i}^{-1}} } \right] \to \operatorname{Spec}R\). These form an open cover if \(\left\langle{f_i}\right\rangle = \left\langle{1}\right\rangle\).
If \(k\in \mathsf{Field}\), there is an equivalence \(\operatorname{Spec}R(k) \cong Z_f(k)\), the zeros of \(f\) in \(k\). Then \(\operatorname{Spec}R \left[ { \scriptstyle { {h}^{-1}} } \right](k) = Z_f(k)\setminus Z_h(k)\) for \(R = {\mathbb{Z}}[x_1,\cdots, x_n]/\left\langle{f}\right\rangle\).
Analog of 2-dimensional \({\mathbb{C}}{\hbox{-}}\)module over a ringer ring: the free \(R{\hbox{-}}\)module \(R{ {}^{ \scriptscriptstyle\times^{2} } }\) of rank 2.
\({\mathbb{P}}^1_{{\mathbb{Z}}}: \mathsf{CRing}\to{\mathsf{Set}}\) is the functor sending \(R\) to the set of direct summands \(M \leq R{ {}^{ \scriptscriptstyle\times^{2} } }\) for which there’s an open covering corresponding to \(\left\{{h_i}\right\}\) where \(M \left[ { \scriptstyle { {h_i}^{-1}} } \right] = M\otimes_R R \left[ { \scriptstyle { {h_i}^{-1}} } \right]\) is a free \(R{\hbox{-}}\)module of rank 1 for all \(i\).
For \(S\in{\mathsf{Alg}_{/R}}\), we have \(\alpha: R\to S\) and for \(N\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}}\) we can forget the module structure along this map by defining \begin{align*} R\times N &\to N \\ (r, n) &\mapsto \alpha(r) \cdot n .\end{align*} This induces a \begin{align*}\[restriction functor\end{align*} ] \(\operatorname{res}_{\alpha}: {\mathsf{S}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).
Conversely we can tensor \(R{\hbox{-}}\)modules up to \(S{\hbox{-}}\)modules to get a functor \(S\otimes_R({-})\), where the interesting bit is \(s\otimes(rm) \mathrel{\vcenter{:}}=\alpha(r) (s\otimes m) = (\alpha(r)s)\otimes m\).
This yields an adjunction: \begin{align*} \adjunction{({-})\otimes_R S}{\operatorname{res}_{\alpha}}{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}} .\end{align*}
Any reasonable property of modules should be preserved by base change!
Descent for modules: when does \(M\otimes_R S\) having property \(P\) as an \(S{\hbox{-}}\)module descend to \(M\) having property \(P\) has an \(R{\hbox{-}}\)module?
Left adjoints are right exact (LARE). In particular, \begin{align*}\[base change\end{align*} ] is right exact, but not always left exact: take \(\alpha: {\mathbb{Z}}\to {\mathbb{Z}}/2\), take the SES \(0 \to {\mathbb{Z}}\xrightarrow{2} {\mathbb{Z}}\to {\mathbb{Z}}/2\to 0\), and tensor with \({\mathbb{Z}}/2\). So an \(R{\hbox{-}}\)algebra \(S\) is flat precisely when the base change \(S\otimes_R({-})\) is exact.
\(S\) is \begin{align*}\[faithfully flat\end{align*} ] when \(S\otimes_R M = 0\implies M=0\). Allows checking things after base-changing to \(S\): - Exactness of any sequence, so in particular injectivity/surjectivity - Finite generation (over \(R\) vs \(S\)) - Projectivity, - Flatness - If \(R\to S\) is faithfully flat and \(R\to T\) is an arbitrary ring morphism, the co-base change \(T\to S\otimes_R T\) is faithfully flat.
General idea: \(R{\hbox{-}}\)modules \(M\) can be specified by \(S\otimes_R M\) along with \begin{align*}\[descent data\end{align*} ].
\begin{align*}\[Faithfully flat descent\end{align*} ] : there is an equivalence of categories \({\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)\),
Given \(F\in [\mathsf{A}, \mathsf{B}]\) and \(G\in [\mathsf{A}, \mathsf{C}]\), the left \begin{align*}\[Kan extension\end{align*} ] of \(G\) along \(F\) is a functor \(L\in [B, C]\) and a sufficiently universal natural transformation \(\alpha\in [G, LF]\).
Tags: #quick_notes
Tags: #terms_and_questions
Tags: #quick_notes
Tags: #reading_notes #derived #infinity_cats
Derived AG: https://people.math.harvard.edu/~lurie/papers/DAG-X.pdf
\begin{align*}\[dg Lie algebras\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-03-49.png\end{align*} ]
\begin{align*}\[elliptic curve\|elliptic curve\end{align*} ] and \begin{align*}\[deformation theory\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-05-28.png\end{align*} ]
\begin{align*}\[presentable infinity category\end{align*} ]. \begin{align*}\[deformation-obstruction theory\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-08-54.png\end{align*} ]
\begin{align*}\[k-linear category\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-19-40.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-21-36.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-28-30.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-30-48.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-33-46.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-34-14.png\end{align*} ]
Weak weak approximation would imply a positive answer to the \begin{align*}\[inverse Galois problem\end{align*} ].
\begin{align*}\[attachments/2021-10-05_20-02-50.png\end{align*} ]
Tags: #stable_homotopy #physics #summaries
Refs: \begin{align*}\[Elliptic cohomology\end{align*} ], \begin{align*}\[Thom-Dold\end{align*} ], \begin{align*}\[Orientability of spectra\|orientability\end{align*} ], \begin{align*}\[formal group law\end{align*} ], \begin{align*}\[ring spectra\end{align*} ], \begin{align*}\[Bousfield localization\end{align*} ], \begin{align*}\[Topological modular forms\|tmf\end{align*} ],
Reference: M-theory, type IIA superstrings, and elliptic cohomology https://arxiv.org/pdf/hep-th/0404013.pdf
\begin{align*}\[attachments/2021-10-05_20-39-39.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-40-20.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-41-16.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-41-33.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-41-56.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-42-42.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-43-37.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-44-09.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-44-36.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-45-25.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-46-47.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-48-43.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-51-54.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-51-38.png\end{align*} ]
Hom is a \begin{align*}\[continuous functor\end{align*} ], i.e. it preserves limits in both variables. Just remember that the first argument is contravariant, so \begin{align*} \cocolim_i \cocolim_j \mathsf{C}(A_i, B_j) = \mathsf{C}(\colim_i A_i, \cocolim_j B_j) .\end{align*}
\begin{align*}\[tannaka duality\end{align*} ] and \begin{align*}\[tannaka reconstruction\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_23-01-03.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_23-04-52.png\end{align*} ]
How \begin{align*}\[K-Theory\end{align*} ] goes:
Minor aside: \({\mathbf{B}}\mathsf{C} \mathrel{\vcenter{:}}={ {\left\lvert {{ \mathcal{N}({\mathsf{C}}) }} \right\rvert} }\).
Start with the category of \begin{align*}\[elliptic curve\|elliptic curves\end{align*} ] : should be pointed \begin{align*}\[algebraic group\end{align*} ], so a \begin{align*}\[slice category\|coslice category\end{align*} ] over a terminal object..?
Refs: \begin{align*}\[algebra valued differential forms\end{align*} ]
https://www.wikiwand.com/en/Lie_algebra-valued_differential_form
\begin{align*}\[Siegel modular forms\end{align*} ] arise as vector-valued differential forms on Siegel modular varieties? See the following paper for leads: https://arxiv.org/abs/math/0605346
\begin{align*}\[adS correspondence conjecture\end{align*} ] :
Tags: #stable_homotopy
Producing a LES:
Integration pairing: for \(E \in {\mathsf{SHC}}(\mathsf{Ring})\), \begin{align*} E^*X &\longrightarrow E_* X \\ \omega \in [\mathop{\mathrm{{\Sigma_+^\infty}}}X, E] &\longrightarrow\alpha \in [{\mathbb{S}}, E\wedge X] \\ \\ {\mathbb{S}}\xrightarrow{\alpha} E \wedge X \cong E\wedge{\mathbb{S}}\wedge X &\cong E \wedge\mathop{\mathrm{{\Sigma_+^\infty}}}X \xrightarrow{1\wedge\omega } E{ {}^{ \scriptscriptstyle\wedge^{2} } } \xrightarrow{\mu} E .\end{align*}
Tags: #category_theory #simplicial #infinity_cats
Recall \({\mathsf{sSet}}= [\Delta^{\operatorname{op}}, {\mathsf{Set}}] = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{Set}}) = {\mathsf{Set}}^{\Delta^{\operatorname{op}}}\).
For \(x_0 \in \mathsf{C}\), a cone from \(x_0\) to \(F\in [J, C]\) for \(J\) any diagram category is a family \(\psi_x: x_0 \to F(x)\) making diagrams commute:
Free \begin{align*}\[cocompletion\end{align*} ] of a category: \(\mathsf{C} \mapsto [\mathsf{C}, {\mathsf{Set}}]\).
Cauchy completeness for a category: closure under all \begin{align*}\[colimits\end{align*} ] that are preserved by every functor.
\begin{align*}\[Subfunctor\end{align*} ] : \(G\leq F\) iff \(G(x) \subseteq F(x)\) and for all \(x \xrightarrow{f} y\), require \(F(f)(G(x)) \subseteq G(y)\).
References: https://arxiv.org/pdf/0801.3480.pdf and https://people.math.umass.edu/~gwilliam/thesis.pdf
Tags: #reading_notes #lie_algebras
String structures on \(X\): spin structures on \({\Omega}X\).
Defining algebra-valued forms when curvature doesn’t vanish:
See \begin{align*}\[factorization algebra\end{align*} ]
\begin{align*}\[derived infinity category\end{align*} ] : \begin{align*}\[differential graded nerve\end{align*} ] of subcategory of \begin{align*}\[fibrant objects\end{align*} ]. Always a \begin{align*}\[stable infinity category\end{align*} ], and localizes at weak equivalences.
Alternatively: take subcategory of fibrant objects, observe \begin{align*}\[enrichment\end{align*} ] over chain complexes, apply \begin{align*}\[Dold-Kan\end{align*} ] to get a simplicial enrichment, then take the \begin{align*}\[homotopy coherent nerve\end{align*} ] or \begin{align*}\[simplicial nerve\end{align*} ].
Getting a chain complex from a \begin{align*}\[simplicial set\end{align*} ] : take free \({\mathbb{Z}}{\hbox{-}}\)modules levelwise, then apply \begin{align*}\[Dold-Kan\end{align*} ].
How (I think?) Postnikov and Whitehead towers are related:
Tags: #idle_thoughts
Idk this weird thing
Tags: #terms_and_questions
Tags: #terms_and_questions
https://math.stanford.edu/~conrad/papers/hypercover.pdf
Tags: #idle_thoughts #simplicial
Not sure how to get this to work yet, but here’s the condition for a functor to be a sheaf:
But we can write \(n{\hbox{-}}\)fold intersections as \begin{align*}\[fiber products\end{align*} ] :
So the condition of \({\mathcal{F}}\) being a sheaf seems to look like letting \({\mathcal{U}}\rightrightarrows X\) be an open cover, setting \(M = {\textstyle\coprod}U_i\), then applying a \begin{align*}\[bar construction\end{align*} ] \begin{align*} M: M{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{1} } } \leftarrow M{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } } \leftarrow\cdots .\end{align*} Then apply \({\mathcal{F}}\), and look at some kind of image sequence? And ask for exactness for \(n\) many levels to get a sheaf, \begin{align*}\[stack\end{align*} ], etc:
The problem is that I don’t really know how to relate the bottom line (whose exactness is the usual condition for sheaves, stacks, etc) to the intermediate steps. This seems like it wants \({\mathcal{F}}({\textstyle\coprod}{-}) = \prod {\mathcal{F}}({-})\), so it commutes with (co?)limits, since probably contravariant functors send coproducts to products. Moreover the bar construction in the 2nd line might form a simplicial object? And the condition of satisfying \begin{align*}\[descent\end{align*} ] is maybe related to either this being a \begin{align*}\[simplicial object\end{align*} ], or its image in the bottom line assembling to a simplicial object, since there are clear degeneracy maps and one would want sections in order to build face maps. Super vague, there are a lot of details missing here!!
Tags: #category_theory
\begin{align*}\[2-category\end{align*} ] : a category \begin{align*}\[Enriched category\|enriched\end{align*} ] in small categories, so hom sets are categories and compositions form bifunctors.
\begin{align*}\[classifying space\end{align*} ] of a 2-cat: replace morphism cats \(C(x, y)\) with \({\mathbf{B}}C(x, y)\) to get a topological 1-cat, then take \({\mathbf{B}}C \mathrel{\vcenter{:}}={ {\left\lvert {{ \mathcal{N}({C}) }} \right\rvert} }\).
For \(F:\mathsf{C}\to \mathsf{D}\), fixing \(p\in D\), can form a \begin{align*}\[homotopy fiber\end{align*} ] 2-category \(y//F\).
Then \({\mathbf{B}}F: {\mathbf{B}}C\to {\mathbf{B}}D\) is a homotopy equivalence of spaces if \(B(y//F) \simeq{\operatorname{pt}}\) is contractible for all \(y\in \mathsf{D}\).
\begin{align*}\[homotopy fiber\end{align*} ] cat: \(y//F\) is a lax \begin{align*}\[comma category\end{align*} ].
\begin{align*}\[lax functor\end{align*} ] :
Tags: #representation_theory #terms_and_questions #number_theory #langlands
\begin{align*}\[Cartan\end{align*} ] subgroup
\begin{align*}\[Borel\end{align*} ] subgroup
\begin{align*}\[Parabolic\end{align*} ] subgroup
\begin{align*}\[local fields\end{align*} ]
\begin{align*}\[valuation\end{align*} ] : For \(v: k \to G\cup\left\{{\infty}\right\}\) and \(G\in {\mathsf{Ab}}\) \begin{align*}\[totally ordered\end{align*} ].
\begin{align*}\[global field\end{align*} ] : algebraic number fields, function fields of \begin{align*}\[algebraic curve\|algebraic curves\end{align*} ] over finite fields (so finite extensions of \({\mathbb{F}}_q { \left( {(} \right) } t))\).
Note the closed point of \(\operatorname{Spec}{ {\mathbb{Z}}_p }\) is \({\mathbb{F}}_p\) and the generic point is \({ {\mathbb{Q}}_p }\).
\begin{align*}\[nonarchimedean field\end{align*} ]
\begin{align*}\[proper morphism\end{align*} ]
\begin{align*}\[Iwahori\end{align*} ] subgroup
Fun fact: \(p{\hbox{-}}\)torsion in an \begin{align*}\[ideal class group\end{align*} ] was the main obstruction to a direct proof of FLT. Observed by Kummer.
Main conjecture of \begin{align*}\[Iwasawa theory\end{align*} ] : two methods of defining \(p{\hbox{-}}\)adic \(L{\hbox{-}}\)functions should coincide. Proved by Mazur/Wiles for \({\mathbb{Q}}\), all totally real number fields by Wiles.
Actual definition of Dirichlet characters:
Define: \begin{align*}\[geometric fiber\end{align*} ]
Reductive, semisimple, simply connected, etc for \(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {S}}\): affine and smooth over \(S\), where geometric fibers are reductive. s.s., etc algebraic groups.
\begin{align*}\[etale morphism\end{align*} ]
Central extension
Fiber functor
Algebraic fundamental group
Certain groups that become isomorphic after field extensions have related automorphic representations.
Langlands dual: \({\mathcal{L}}(G)\) controls \({\mathsf{G}{\hbox{-}}\mathsf{Mod}}\) somehow, arises as an extension \({ \mathsf{Gal}} (k^s _{/ {k}} ) \to {\mathcal{L}}(G) \to H\) where \(H \in \operatorname{Lie}{\mathsf{Grp}}_{/ {{\mathbb{C}}}}\).
A connected reductive algebraic group over a separably closed field \(k\) is uniquely determined by its root datum.
\begin{align*}\[Langlands dual\end{align*} ] : take root datum, dualize datum, take associated group.
Langlands’ strategy for proving local and global conjectures: \begin{align*}\[Arthur-Selberg trace formula\end{align*} ].
Equivalence of orbital integrals can somehow be related to \begin{align*}\[Springer Fiber\end{align*} ]??
Starting point for Langlands: \begin{align*}\[Artin reciprocity\end{align*} ], generalizing \begin{align*}\[quadratic reciprocity\end{align*} ].
\begin{align*}\[Chebotarev density\end{align*} ] theorem is a generalization of \begin{align*}\[Dirichlet's theorem on arithmetic progressions\end{align*} ].
“The Langlands conjectures associate an automorphic representation of the adelic group \(\operatorname{GL}_n({\mathbb{A}}_{/ {{\mathbb{Q}}}} )\) to every \(n{\hbox{-}}\)dimensional irreducible representation of the Galois group, which is a \begin{align*}\[cuspidal representation\end{align*} ] if the Galois representation is irreducible, such that the \begin{align*}\[Artin L function\end{align*} ] of the Galois representation is the same as the \begin{align*}\[automorphic L function\end{align*} ] of the automorphic representation”
Serre’s modularity conjecture: an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form
\({\mathbb{A}}_{/ {{\mathbb{Q}}}}\): keeps track of all of the completions of \({\mathbb{Q}}\) simultaneously.
Reciprocity conjecture: a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an \(L\)-group
\begin{align*}\[geometric Langlands\end{align*} ] : relates \begin{align*}[l-adic representations](#l-adic-representations)\end{align*} of the \begin{align*}\[étale fundamental group\end{align*} ] of an \begin{align*}\[algebraic curve\end{align*} ] to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.
2018: Lafforgue established \begin{align*}\[Global Langlands correspondence\|global Langlands\end{align*} ] for automorphic forms to \begin{align*}\[Galois representations\end{align*} ] for connected reductive groups over global function fields
\begin{align*}\[purity\end{align*} ] : happens in a specific codimension
Tags: #homological_stability #representation_theory
Reference: Church-Ellenberg-Farb
Smallest subrings of \begin{align*}\[Chow ring\|Chow\end{align*} ] closed under \begin{align*}\[pushforward\end{align*} ] by forgetful/gluing maps between various \({ \mathcal{M}_{g, n} }\).
Can push through the \begin{align*}\[cycle class map\end{align*} ], unknown these are isomorphic.
Can get a surjection \({\mathbb{Q}}[\kappa_i] \to H^*({ \mathcal{M}_{g, n} })\) for degree high enough.
Main result: dimensions of representation stabilize.
Sequence of \(S_n{\hbox{-}}\)reps converted into a single \({\mathsf{FI}}{\hbox{-}}\)module.
\({\mathsf{{\mathsf{FI}}}{\hbox{-}}\mathsf{Mod}} \mathrel{\vcenter{:}}= F\in {\mathsf{Fun}}({\mathsf{FI}}, {\mathsf{k}{\hbox{-}}\mathsf{Mod}})\).
Gradings: functors from \({\mathbb{N}}\to {\mathsf{k}{\hbox{-}}\mathsf{Mod}}\).
Tags: #modular_forms #moduli_spaces #stacks
Tags: #reading_notes Refs: \begin{align*}\[modular form\end{align*} ]
Reference: see https://www.math.purdue.edu/~arapura/preprints/shimura2.pdf
\({\operatorname{SL}}_2({\mathbb{R}})\curvearrowright{\mathbb{H}}\) transitively by linear fractional transformations, and \({\operatorname{Stab}}(i) = {\operatorname{SO}}(2)\). Thus one can realize \({\mathbb{H}}\cong {\operatorname{SL}}_2({\mathbb{R}})/{\operatorname{SO}}_2({\mathbb{R}})\).
Applying a homothety to a \begin{align*}\[lattice\end{align*} ] \(\Lambda\) yields \(L_\tau \mathrel{\vcenter{:}}={\mathbb{Z}}+ {\mathbb{Z}}\tau\) for some \(\tau\in{\mathbb{H}}\) and \(\Lambda \cong L_\tau\). Writing an elliptic curve as \({\mathbb{C}}/L_\tau\), the moduli of elliptic curves is given by \begin{align*} A_1\mathrel{\vcenter{:}}={\operatorname{SL}}_2({\mathbb{Z}})\diagdown{\mathbb{H}}\cong \dcoset{{\operatorname{SL}}_2({\mathbb{R}})}{{\operatorname{SL}}_2({\mathbb{Z}})}{{\operatorname{SO}}_2({\mathbb{R}})} .\end{align*} This quotient is Hausdorff, and \(A_1 \xrightarrow{\sim} {\mathbb{C}}\) as topological spaces. Somehow this comes from “gluing the two bounding lines of \(F\) and folding the circular boundary in half,” yielding the sphere minus a point.
\(-I\) acts trivially on \({\mathbb{H}}\), so this factors through \(\Gamma \mathrel{\vcenter{:}}={\operatorname{PSL}}_2({\mathbb{Z}}) \mathrel{\vcenter{:}}={\operatorname{SL}}_2({\mathbb{Z}})/\left\langle{\pm I}\right\rangle\).
Letting \(S= (z\mapsto -1/z) = \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right), T = (z\mapsto z+1) = \left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)\), there are fundamental domains:
\(i\) has \begin{align*}\[isotropy\end{align*} ] \(\left\langle{S}\right\rangle\), \(\zeta_3\) has \(\left\langle{ST}\right\rangle\), and \(\zeta_3^2\) has \(\left\langle{TS}\right\rangle\). Applying \(S\) and \(T^{\pm 1}\) to the fundamental domain \(F\) tiles \({\mathbb{H}}\) by hyperbolic triangles.
\({\operatorname{PSL}}_2({\mathbb{Z}}) = \left\langle{S, T}\right\rangle\).
Maps \(f: A_1\to {\mathbb{C}}\) are continuous iff their pullbacks along \(\pi: {\mathbb{H}}\to A^1\) are continuous, so these are necessarily \(\Gamma{\hbox{-}}\)invariant functions.
\(f\) is \begin{align*}\[automorphic\end{align*} ] with automorphy factor \(\phi_\gamma(z)\) iff \begin{align*} f(\gamma z) = \phi_\gamma(z) f(z) .\end{align*} For any two such functions, their ratio \(g=f_1/f_2\) satisfies \(g(\gamma z) = g(z)\).
\(f\) is weakly modular of weight \(2k\) if \begin{align*} f(z) = (cz+d)^{-2k} f(\gamma z), \gamma \mathrel{\vcenter{:}}={ \begin{bmatrix} {a} & {b} \\ {c} & {d} \end{bmatrix} } .\end{align*}
Note that \begin{align*} {\frac{\partial }{\partial z}\,} {az+b \over cz +d }= {ad-bc \over (cz+d)^2} = {1\over (cz+d)^2} ,\end{align*} and so a meromorphic form \(\omega = f(z) \,dz\) transforms under \(\gamma\) as \begin{align*} \gamma \cdot f(z)\,dz= f(\gamma \cdot z) d(\gamma\cdot z) = (cz+d)^{-2}f(\gamma\cdot z)\,dz .\end{align*}
So weakly modular forms of weight \(2k\) are those form which \(\omega^{\otimes k}\) is invariant.
Dropping the weakly adjective involves imposing holomorphy conditions at \(\infty\). \(f\) is a (standard) \begin{align*}\[modular form\end{align*} ] of \begin{align*}\[weight of a modular form\|weight\end{align*} ] \(2k\) if \(f\) is weakly modular, holomorphic on \({\mathbb{H}}\), and the Fourier coefficients satisfy \(a_{<0} = 0\).
Poisson summation: if \(f:{\mathbb{R}}\to {\mathbb{C}}\) is a \begin{align*}\[Schwartz function\end{align*} ] (smooth and super-polynomial decay), then \begin{align*} \sum_{n \in \mathbb{Z}} f(n)=\sum_{n \in \mathbb{Z}} \widehat{f}(n) .\end{align*}
\(\zeta(s)\) is the \(L{\hbox{-}}\)function associated to the trivial Galois representation \begin{align*} \rho_{\mathop{\mathrm{Triv}}}: G_{\mathbb{Q}}\to {\mathbb{C}}^{\times} .\end{align*} \(L{\hbox{-}}\)functions coming from arbitrary 1-dim reps will correspond to Dirichlet characters by Kronecker-Weber, and are referred to as Dirichlet \(L{\hbox{-}}\)functions.
\(\Gamma(1) \mathrel{\vcenter{:}}={\operatorname{SL}}_2({\mathbb{Z}})\), and principal congruence subgroups of level \(N\) for \(\Gamma(1)\) are defined as \begin{align*} \Gamma(N) \mathrel{\vcenter{:}}=\ker\qty{\Gamma(1) \twoheadrightarrow{\operatorname{SL}}_2({\mathbb{Z}}/N)} = \left\{{M\in \Gamma(1) {~\mathrel{\Big\vert}~}M\cong I \operatorname{mod}N}\right\} ,\end{align*} so the kernels of reduction mod \(N\). \begin{align*}\[Congruence subgroups\end{align*} ] are any subgroups \(H\) such that \(\Gamma(N) \subseteq H \leq \Gamma(1)\) for some \(N\).
Letting \(\Gamma(N)\) act on \({\mathbb{H}}\) or \({\mathbb{H}}^*\), one can define \begin{align*}\[modular curves\end{align*} ] \begin{align*} X(\Gamma) &\mathrel{\vcenter{:}}=\Gamma\diagdown {\mathbb{H}}^* \\ Y(\Gamma) &\mathrel{\vcenter{:}}=\Gamma\diagdown {\mathbb{H}} ,\end{align*} where \({\mathbb{H}}^* = {\mathbb{H}}\cup({\mathbb{Q}}\cup\left\{{\infty}\right\}) \subset {\mathbb{P}}^1({\mathbb{C}})\).
The inclusions \(\Gamma \hookrightarrow\Gamma(1)\) induce a branched cover \(X(\Gamma) \twoheadrightarrow X(\Gamma(1)) = A^1 \cong {\mathbb{P}}^1({\mathbb{C}})\).
The genera of these curves can be computed using \begin{align*}\[Riemann-Hurwitz\end{align*} ] : \begin{align*} 2 g(Y)-2=(2 g(X)-2) d+\sum_{y \in Y}\left(e_{y}-1\right) ,\end{align*} yielding for \(N\geq 3\), \(g(X(\Gamma(N)))\) is given by \begin{align*} g=1+\frac{d(N-6)}{12 N} {\quad \operatorname{where} \quad} d=\frac{1}{2}[\Gamma(1): \Gamma(N)]=\frac{N^{3}}{2} \prod\left(1-\frac{1}{p^{2}}\right) .\end{align*}
For \(X\) a smooth curve and \(D\in \operatorname{Div}(X)\), set \begin{align*} \Omega^1_X(D) \mathrel{\vcenter{:}}=\Omega^1_X \otimes{\mathcal{O}}_X(D) \cong {\mathcal{O}}_X(\omega + D) \end{align*} where \(\omega\) is the canonical divisor. Then \({{\Gamma}\qty{X; \Omega^1_X(D)} }\) is the space of meromorphic 1-forms \(\omega\) such that \(\operatorname{Div}(\omega) + D \geq 0\) is effective.
Define \(M_{2k}(\Gamma)\) to be the space of weight \(2k\) modular forms, and \(S_{2k}\) the space of cusp forms. Then \(\bigoplus_k S_{2k} \in {\mathsf{gr}\,}^{\mathbb{Z}}{\mathsf{Alg}}_{/ {{\mathbb{C}}}} ^{\mathrm{fg}}\), and for \({\operatorname{SL}}_2({\mathbb{Z}})\) this algebra is generated by the Eisenstein series \(G_4\) and \(G_6\).
A contravariant functor \(F\) admits a \begin{align*}\[fine moduli space\end{align*} ] \({\mathbf{B}}F\) if \(F\) is representable by \({\mathbf{B}}F\), i.e. \(F({-}) \cong \mathop{\mathrm{Hom}}({-}, {\mathbf{B}}F)\). By Yoneda, \(F\) admits a universal family \({\mathbf{E}}F \to {\mathbf{B}}F\) so that \(F(X)\) is the pullback of it under some map \(X\to {\mathbf{B}}F\).
The functor \(F({-})\) sending \(X\) to isomorphism classes of elliptic curves over \(X\) admits \(Y(1)\) as a \begin{align*}\[fine moduli space\|coarse moduli space\end{align*} ] and not a fine one, since there are nontrivial families with constant \(j{\hbox{-}}\)invariant. Despite this, \(E\to j(E)\) gives a bijection between isomorphism classes of elliptic curves and points of \(Y(1)\).
A level \(N\) structure is a basis for \(H_1(E; {\mathbb{Z}}/N)\), which is symplectic since it carries a pairing with intersection matrix \({ \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} }\).
Moduli interpretations:
An \begin{align*}\[elliptic curve\end{align*} ] \(E\) over a \begin{align*}\[scheme\end{align*} ] \(S\) is a \begin{align*}\[smooth proper morphism\end{align*} ] \(f:E\to S\) with a section such that the closed fibers of \(f\) are genus 1 curves.
Look up the \begin{align*}\[Weil Pairing\end{align*} ] \(e_n\).
A level \(N\) structure is a pair of points \(P, Q \in E[N]\) generating a subgroup with that \(e_n(P, Q) = \zeta_N\) is a primitive \(N\)th root of unity. More generally, for curves over schemes, this is a pair of sections inducing level structures on closed fibers.
For \(N=2\), \(Y(2) = \operatorname{Spec}{\mathbb{Z}} { \left[ {t, {1\over t(t-1)}} \right] }\) as a coarse moduli space, and a corresponding almost-universal family \(y^2z = x(x-z)(x-tz)\).
For \(N=3\), \(Y(2) = \operatorname{Spec}R { \left[ {t, {1\over t^3-1}} \right] }\) where \(R \mathrel{\vcenter{:}}={\mathbb{Z}} { \left[ {{1\over 3}, \zeta_3^2} \right] }\) for \(\zeta_3\) a primitive third root of unity. The universal family is \(x^3 + y^3 + z^3 = 3txyz\), where the level 3 structure is given by the sections \({\left[ {-1, 0, 1} \right]}, [-1, \zeta_3^2, 0]\).
Can view an elliptic curve as a pair \((X, p)\) where \(X\) is a compact Riemann surface with \(\dim_{\mathbb{C}}H^0(X; \Omega^1_X) = 1\) and \(p\) is a point.
Can define a lattice \(\Lambda \subseteq V\) in an arbitrary vector space as a discrete \begin{align*}\[cocompact\end{align*} ] subgroup, so \(V/\Lambda\) is compact.
The order of a function \(f\) at \(x\) is given by
\begin{align*} \text { ord }_{x}(f):= \begin{cases}0 & \text { if } \mathrm{f} \text { is holomorphic and non-zero at } x \\ k & \text { if } \mathrm{f} \text { has a zero of order } k \text { at } x \\ -k & \text { if } \mathrm{f} \text { has a pole of order } k \text { at } x\end{cases} .\end{align*}
Can define divisors as maps \(D:X\to {\mathbb{Z}}\) where cofinitely many points are sent to zero. The map \({\operatorname{Ord}}_{{-}}(f)\) is a divisor associated to any function \(f\), denoted \((f)\).
Divisors \((f)\) for \(f\) meromorphic are principal, and setting \(\deg(\sum n_i p_i) \mathrel{\vcenter{:}}=\sum n_o\), it turns out that \(\deg((f)) = 0\) for \(f\) principal.
The period map is defined as \begin{align*} \Phi: H_{1}(X, \mathbb{Z}) & \rightarrow \mathbb{C} \\ \gamma & \mapsto \int_{\gamma} \omega .\end{align*} For a fixed nonzero holomorphic 1-form \(\omega\), there is a group pf \begin{align*}\[periods\end{align*} ] which forms a lattice over \({\mathbb{C}}\): \begin{align*} \Lambda \mathrel{\vcenter{:}}=\left\{{\int_\gamma \omega \in {\mathbb{C}}{~\mathrel{\Big\vert}~}\gamma \in H_1(X; {\mathbb{Z}})}\right\} = \operatorname{im}\Phi .\end{align*} One can recover \((X, P)\) as \(({\mathbb{C}}/\Lambda(\omega), 0)\)
To pick back up: https://repositorio.uniandes.edu.co/bitstream/handle/1992/43725/u830743.pdf?sequence=1
Tags: #category_theory #homotopy_theory
\begin{align*}\[mapping cone\end{align*} ] as a \begin{align*}\[pushout\end{align*} ] and \begin{align*}\[mapping fiber\end{align*} ] as \begin{align*}\[pullback\end{align*} ] :
Tags: #probability #review_material #undergraduate
Tags: #combinatorics #review_material #undergraduate
Prop: the number of \begin{align*}\[integer compositions\end{align*} ] of \(n\) into exactly \(k\) parts is \({n-1 \choose k-1}\), so \(\sum a_i = n\) with \(a_i\geq 1\) for all \(i\).
Prop: the number of \begin{align*}\[compositions\end{align*} ] of \(n\) into any number of parts is \(2^{n-1}\).
Prop: the number of weak compositions of \(n\) into exactly \(k\) parts is \({n+k-1 \choose k-1}\).
Quotient set of compositions by permutations to get \begin{align*}\[integer partitions\end{align*} ].
\begin{align*}\[Stirling numbers\end{align*} ] : the number of ways to partition an \(n\) element set into exactly \(k\) nonempty unlabeled disjoint blocks whose disjoint union is the original set.
Tags: #stable_homotopy #k_theory
\(K^\mathsf{Alg}({\mathbb{S}})\) is of interest due to Waldhausen’s stable parameterized \begin{align*}\[h-cobordism theorem\end{align*} ].
A first approximation: \({\mathsf{K}}^\mathsf{Alg}(L_{K(n)} {\mathbb{S}})\), so localize with respect to \begin{align*}\[Morava K-theory\end{align*} ]
Rognes extends \begin{align*}\[Lichtenbaum-Quillen conjectures\end{align*} ] and develops a theory for Galois extensions of \({\mathbb{S}}{\hbox{-}}\)algebras
\begin{align*}\[Redshift\end{align*} ] : \begin{align*}\[algebraic K-theory\end{align*} ] increases \begin{align*}\[chromatic complexity\end{align*} ] by one.
Refs: \begin{align*}\[A1 Homotopy\end{align*} ]
Context: the infinity category of spaces, i.e. \begin{align*}\[homotopy types\end{align*} ]
Take smooth manifolds, take the \begin{align*}\[Yoneda embedding\end{align*} ] to \(\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Mfd}})\): these satisfy a Mayer-Vietoris gluing property, and homotopy invariance in the sense that \begin{align*} \mathop{\mathrm{Hom}}({-}, X) \cong \mathop{\mathrm{Hom}}({-}\times I, X) .\end{align*}
AG setting: \(\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Sch}}_{/k})\), send to presheaves to define \begin{align*}\[motivic spaces\end{align*} ].
See \begin{align*}\[Betti realization\end{align*} ] for \(k={\mathbb{C}}\): \({\mathsf{sm}}{\mathsf{Sch}}_{/{\mathbb{C}}}\to {\mathsf{Spaces}}\) where \(X\mapsto X({\mathbb{C}})\).
From topology: identify \({\mathbf{B}}{\operatorname{U}}_n({\mathbb{C}}) = {\operatorname{Gr}}_n({\mathbb{C}})\) to get \begin{align*} { \mathsf{Vect} }_{/{\mathbb{C}}}^{\operatorname{rank}= n}(U) \cong \pi_0 \mathop{\mathrm{Maps}}(U, {\mathbf{B}}{\operatorname{U}}_n({\mathbb{C}})) .\end{align*}
Problem in AG: there are two rank 2 \begin{align*}\[vector bundles\end{align*} ] on \({\mathbb{P}}^1 \times{\mathbb{A}}^1\) whose fibers over 0 and 1 are \({\mathcal{O}}^2\) and \({\mathcal{O}}(1) \oplus {\mathcal{O}}(-1)\).
Theorem: for \(U\) smooth affine \({\mathsf{Sch}_{/k}}\), there is an equivalence of rank \(n\) vector bundles on \(U\) mod equivalence to \(\pi_0 \mathop{\mathrm{Maps}}_{{\mathsf{Spaces}}(k)}(U, {\mathbf{B}}\operatorname{GL}_n)\) where again \({\mathbf{B}}\operatorname{GL}_n \cong {\operatorname{Gr}}_n\).
\begin{align*}\[Algebraic K theory\end{align*} ] : finitely generated projective \(R{\hbox{-}}\)modules mod equivalence with \(\oplus\), then take group completion to get \(K_0(R)\).
To get a space: take \(K(R) \mathrel{\vcenter{:}}=\mathop{\mathrm{Proj}}({\mathsf{R}{\hbox{-}}\mathsf{Mod}})^ {\operatorname{gp} }\) to get a space, set \(K_i R \mathrel{\vcenter{:}}=\pi_i K(R)\)
Prop: the \(K\) theory space here is a motivic space, \(K: {\mathsf{sm}}{\mathsf{Sch}}_{/k}^{\operatorname{op}}\to {\mathsf{Spaces}}\).
Interesting fact: \({\Omega}^\infty {\mathbb{S}}\cong ({\mathsf{FinSet}}, {\textstyle\coprod})^ {\operatorname{gp} }\).
Note from Yuri Sulyma: "B is (widespread but) really bad notation for geometric realization. You should think of B as part of an equivalence \begin{align*} {\mathbf{B}}: \left\{{\text{monoidal categories}}\right\} \to \left\{{\text{pointed connected (2-)categories}}\right\} \end{align*}
up to the \begin{align*}\[Quillen equivalence\end{align*} ]
\begin{align*} {\mathsf{Kan}}\xrightarrow{\sim} {\mathsf{Spaces}} \end{align*}
\begin{align*}\[geometric realization\end{align*} ] takes a (quasi-)category (or \begin{align*}\[simplicial set\end{align*} ]) and inverts all the morphisms.
So \(M^ {\operatorname{gp} } = \Omega{ {\left\lvert {{\mathbf{B}}M} \right\rvert} }\): you take \(M\), \begin{align*}\[deloop\end{align*} ] to turn the objects into morphisms, invert all the morphisms, then take loops to get your objects back."
Theorem ( \begin{align*}\[Morel-Voevodsky\end{align*} ]): \(X\in {\mathsf{sm}}{\mathsf{Sch}_{/k}}\) \begin{align*} K(X) \cong \mathop{\mathrm{Maps}}_{{\mathsf{Spaces}}(k)}(X, {\mathbb{Z}}\times{\operatorname{Gr}}_\infty) .\end{align*}
Uses \begin{align*}\[stratification\end{align*} ] of \({\operatorname{Gr}}\) by affines, thanks \begin{align*}\[Schubert calculus\end{align*} ]!
There is an \({\mathbb{A}}^1\) homotopy equivalence on affines: \(K \simeq{\mathbb{Z}}\times{\operatorname{Gr}}_\infty\). Also, \begin{align*} {\operatorname{Betti}}(k) \simeq{\mathbb{Z}}\times{\operatorname{BU}}= {\Omega}^\infty{\operatorname{KU}} .\end{align*}
Theorem: Can replace \({\operatorname{Gr}}\) with \(\operatorname{Hilb}_\infty({\mathbb{A}}^\infty)\). Very singular!
Definition: \(\operatorname{Hilb}_d({\mathbb{A}}^n)(T)\) are maps \(Z\hookrightarrow{\mathbb{A}}^n\times T\) over \(T\) which are finite \begin{align*}\[flat morphism\|flat\end{align*} ] of degree \(d\) over \(T\).
Morally: \(d{\hbox{-}}\)tuples of points in \({\mathbb{A}}^n\).
Representable! But \(\operatorname{Hilb}_\infty\) is a colimit, thus an \begin{align*}\[Ind scheme\end{align*} ].
This says either the \begin{align*}\[Hilbert scheme\end{align*} ] or \begin{align*}\[K-Theory\end{align*} ] is hard.
In fact the theorem defines a map \({\operatorname{Gr}}_{d-1} \to \operatorname{Hilb}_d({\mathbb{A}}^\infty)\) sending a vector space to the tangent space at 0, and proves this is an \({\mathbb{A}}^1{\hbox{-}}\)homotopy equivalence on affines.
Sends subspace to thick point at zero.
Burt’s proof worked!
\begin{align*}\[Grassmannian\end{align*} ] : parameterizes vector bundles with an embedding into \(\infty\)?
First step in proof: forget embedding into \({\mathbb{A}}^\infty\), send \({ \mathsf{Vect} }_{d-1}\) to finite flat schemes of degree \(d\) \({\mathsf{FFlat}}_d(R)\) over \(\operatorname{Spec}R\), which are stacks.
Send \(V\to R \oplus V\), a \begin{align*}\[square zero extension\end{align*} ], add trivial multiplication.
Inverse: take an algebra \(A\to A/R\) by killing the unit.
Not an equivalence of stacks! Since \(A\not\cong A/R \oplus R\), but the surprising fact is \(A\to A/R \oplus R\) is \({\mathbb{A}}^1\) homotopic to the identity on \({\mathsf{FFlat}}(R)\).
Cook up an explicit homotopy: take the \begin{align*}\[Rees algebra\end{align*} ] \begin{align*} {\operatorname{Rees}}(A) \mathrel{\vcenter{:}}=\left\{{ a_0 + a_1 t + \cdots {~\mathrel{\Big\vert}~}a_0\in R }\right\} \subseteq A[t] .\end{align*}
\({\operatorname{Rees}}(A) / \left\langle{ t-1 }\right\rangle\simeq A\)
\({\operatorname{Rees}}(A) / \left\langle{ t }\right\rangle\simeq R \oplus A/R\).
\[\[Hermitian K theory\]
\] :
- Use orthogonal Grassmannian, take vector bundles with extra data of nondegenerate symmetric
\[\[bilinear form\]
\]. Need \( \char k \neq 2 \). Take
\[\[Gorenstein\]
\] closed subschemes, which is extra data of orientation.Tags: #stable_homotopy #seminar_notes
Mike was thinking about computing \begin{align*}\[Topological modular forms\|tmf\end{align*} ] at the prime \(p=3\), since for \(p>3\) it breaks up as a wedge of copies \({\operatorname{BP}}\left\langle{ 2 }\right\rangle\) of \begin{align*}\[Brown-Peterson spectra\end{align*} ]
Roughly twice as hard as computing \begin{align*}\[K-Theory\end{align*} ] with \begin{align*}\[ku\end{align*} ]! (Wilson, Adams, Margalis)
For \(p=2\): an \begin{align*}\[Adams spectral sequence\end{align*} ] (Mahowald, Davis-Mahowald) built out of \begin{align*} H^*( \mathrm{tmf} , {\mathbb{F}}_2) \cong A \otimes_{A(2) } {\mathbb{F}}_2 && \text{where } A(2) = \left\langle{ \operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{Sq}^4 }\right\rangle \end{align*}
For \(p=3\), heuristic: should be like \begin{align*}\[ko\end{align*} ] at \(p=2\) in terms of complexity.
Also thinking about Hopkins-Miller higher real K theories.
First Talbot: huge efforts by Norrah!!!
\begin{align*}\[Formal group\|formal group laws\end{align*} ] over \(R\): a power series \(x +_F y \mathrel{\vcenter{:}}= F(x, y) \in R { \left[ {x, y} \right] }\) such that
A morphism of \begin{align*}\[Formal group\|formal group laws\end{align*} ] : \(f\in R { \left[ {x} \right] }\) with \(f(x+_F y)= f(x) +_G f(y)\).
The functor \(R\to \mathsf{FGL}_{/R}\) is representable, as is the functor sending \(R\) to formal group laws over \(R\) along with an isomorphism \(f\) such that \(f'(0) = 1\).
Theorem (Quillen): \({\operatorname{MU}}_*\) is the ring representing the first functor. See \begin{align*}\[MU\end{align*} ].
Milnor showed \({\operatorname{MU}}_* = {\mathbb{Z}}[x_1, \cdots]\).
How to prove representability: take representing object for power series, check what the conditions translate to.
\({\operatorname{MU}}_* {\operatorname{MU}}\) represents the second factor (i.e. the \({\operatorname{MU}}_*\) homology of \({\operatorname{MU}}\), given by \(\pi_*({\operatorname{MU}}\wedge{\operatorname{MU}}))\).
Example: if \(n\in {\mathbb{N}}\), then \begin{align*} [n]_F (x) = \overset{F}{\sum_{k\leq n}} x = nx + \cdots \end{align*} is an endomorphism of \(F\).
If \(\char R = p\), then \([p]_F (x) = f(x^{p^n})\), if \(f'(0) \in R^{\times}\) then the \(\operatorname{ht}F=n\) and \(f(x) = v_n x + \cdots\).
For \(R\) a field, it’s a theorem that the \begin{align*}\[height of a formal group law\end{align*} ] is a complete invariant for algebraically closed fields.
Picture of the \begin{align*}\[moduli of formal group laws\end{align*} ] :
\begin{align*}\[attachments/image_2021-06-06-15-32-17.png\end{align*} ]
How to glue: sheaf condition on opens? Extensions on closed sets? But how do you talk about gluing an open to a closed set?
Explained by \begin{align*}\[deformation theory\end{align*} ] : can push not only in direction in the space, but also into the tangent space directions.
\begin{align*}\[deformation\end{align*} ] : a ring map \(A\to k\) with a nilpotent kernel:
\begin{align*}\[attachments/image_2021-06-06-15-35-33.png\end{align*} ]
\begin{align*}\[attachments/image_2021-06-06-15-37-51.png\end{align*} ]
Here \(\widehat{G}\) sends a ring \(R\) to the set of nilpotent elements of \(R\), and \(F\) gives that a group structure – the algebraic geometry gadget corresponding to the formal group law \(F\).
Examples:
Theorem (Lubin-Tate): there is a universal deformation for \((k, F)\) given by \begin{align*} {\mathbb{W}}(k) { \left[ {{ {u}_1, {u}_2, \cdots, {u}_{n-1}}} \right] } \mathrel{\vcenter{:}}= E(k, F)_0 .\end{align*}
See \begin{align*}\[Lubin-Tate theory\end{align*} ] and \begin{align*}\[Witt Vectors\|Witt vector\end{align*} ].
For \(k = {\mathbb{F}}_p\), we have \({\mathbb{W}}(k) = { {\mathbb{Z}}_p }\), and there is an action of \(\mathop{\mathrm{Aut}}(F)\) and \({ \mathsf{Gal}} (k)\).
Theorem ( \begin{align*}\[Goerss-Hopkins-Miller\end{align*} ]): there is a canonical functor \((k, F) \to E(k, F)\) such that
This lifts the AG problem to a problem in commutative \begin{align*}\[ring spectra\end{align*} ].
Theorem (Devinats-Hopkin?): the map \begin{align*} L_{K(n)} S^0 \xrightarrow{\simeq} E_n^{h{\mathbb{G}}_n} \end{align*} is an equivalence where \begin{align*} E_n &\mathrel{\vcenter{:}}= E({\mathbb{F}}_{p^n}, F_{\mathrm{Honda}}) \\ {\mathbb{G}}_n &\mathrel{\vcenter{:}}={ \mathsf{Gal}} ({\mathbb{F}}_{p^n}) \rtimes S_n \\ S_n &\mathrel{\vcenter{:}}=\mathop{\mathrm{Aut}}(F_{\mathrm{Honda}}) .\end{align*}
Define \begin{align*}\[Hopkins-Miller higher real K theories\end{align*} ] : for \(G \subseteq S_n\) finite, \({\mathsf{E} {\operatorname{O}}}_n(G) = E_n^{hG}\).
Example: for \(n=1, p=2\) we have \({\mathsf{E} {\operatorname{O}}}_1(C_2) = {\operatorname{KO}}{ {}^{ \widehat{2} } }\), which is completely understood via the \begin{align*}\[homotopy fixed point spectral sequence\end{align*} ].
Example: for \(n=2\), we know \begin{align*} {\mathsf{E} {\operatorname{O}}}_2(G) = L_{K(2)} \mathrm{tmf} \end{align*} or a summand thereof.
Use Adams indexing, look at homotopy fixed point spectral sequence \begin{align*} H^s(G; \pi_t E_n)\Rightarrow\pi_{t-s} {\mathsf{E} {\operatorname{O}}}_n(G) .\end{align*}
We know a lot about the bottom row: \(H^0(G; \pi_* E_n) = (\pi_* E_n)^G\), using that \begin{align*}\[group cohomology\end{align*} ] is the derived functor of \(G{\hbox{-}}\)invariants.
If \({\sharp}G\) is prime to \(p\) then \(H^{> 0}(G, \pi* E_n) = 0\).
We don’t know much else about anything in this spectral sequence!
Consider \begin{align*} {\mathcal{O}}_n \mathrel{\vcenter{:}}={\mathbb{W}}({\mathbb{F}_{p^n}}) \left\langle{ s }\right\rangle / \left\langle{ sa = a^{ \varphi} S }\right\rangle \end{align*} where \(\varphi\) is the Frobenius on \({\mathbb{F}_{p^n}}\).
This is the \begin{align*}\[Dieudonne module\end{align*} ] for \(F_\mathrm{\operatorname{Honda}}\)?
Power series rings have a maximal ideal \({\mathfrak{m}}\), so consider \begin{align*} \pi_{-2} E_n / \left\langle{ p, {\mathfrak{m}}^2}\right\rangle .\end{align*}
There is an \({\mathcal{O}}_n^{\times}\) \begin{align*}\[equivariant\end{align*} ] map from \({\mathcal{O}}_n/p\) to this, and the question is how to lift:
\begin{align*}\[attachments/image_2021-06-06-15-58-57.png\end{align*} ]
Devinats-Hopkins: up to \begin{align*}\[associated graded\end{align*} ], \begin{align*} \pi_* E_n \cong \operatorname{Sym}({\mathcal{O}}_n) \left[ { \scriptstyle { {\Delta}^{-1}} } \right]{ {}^{ \widehat{I} } } .\end{align*}
Use that the symmetric algebra is a free thing.
Warning: can’t make this \(S_n\) \begin{align*}\[equivariant\end{align*} ], so can’t compute the whole thing using this approach.
“Up to associated graded” means there’s a spectral sequence relating.
Mike thought about this a lot in grad school! But wound up doing his these on the first topic about \(H^*\) of \begin{align*}\[tmf\end{align*} ].
2007 Talbot: Mike Hopkins was the faculty mentor.
Theorem ( \begin{align*}\[Hill-Hopkins-Ravenel\end{align*} ]): there is a \(G{\hbox{-}}\)equivariant lift \({\mathcal{O}}_n \to \pi_{-2} E_n\) for any finite \(G\).
Real K theories see a lot of \(\pi_* {\mathbb{S}}\).
Theorem (Ravenel): for \(p\geq 5\), some element \(\beta_{p^i / p^i}\) does not survive the \begin{align*}\[ANSS\end{align*} ].
Exact same argument for \begin{align*}\[Kervaire invariant 1\end{align*} ] : \(p=2\) version of this argument?
\({ \underset{(\infty, {2})}{ \mathsf{Cat}} }\): discrete set of objects, enriched in categories, \(2{\hbox{-}}\)morphisms are strictly associative?
\({ \underset{(\infty, {n})}{ \mathsf{Cat}} }\): all \(n+k{\hbox{-}}\)morphisms are invertible.
There is an embedding \(n{\hbox{-}}\mathsf{Cat}\hookrightarrow{ \underset{(\infty, {n})}{ \mathsf{Cat}} }\)with a specific model structure.
Model structure on \({\mathsf{sSet}}\): fibrant objects are Kan complexes, there is a Quillen equivalence between \({\mathsf{Top}}\) and \({\mathsf{sSet}}\)
\({ \underset{(\infty, {1})}{ \mathsf{Cat}} }\): enriched in \({ \underset{\infty}{ \mathsf{Grpd}} }\)
\begin{align*}\[attachments/image_2021-06-05-12-16-29.png\end{align*} ]
\begin{align*}\[Joyal model structure\end{align*} ] : \({\mathsf{sSet}}\) with quasicategories as fibrant objects.
\begin{align*}\[Quillen equivalence\end{align*} ] to the \begin{align*}\[Kan model structure\end{align*} ] by taking the \begin{align*}\[homotopy coherent nerve\end{align*} ]?
Exercise: \begin{align*}\[nerve\end{align*} ] is a \begin{align*}\[Kan complex\end{align*} ] iff \(\mathsf{C}\) is a \begin{align*}\[groupoid\end{align*} ].
Complete \begin{align*}\[Segal space\end{align*} ] : a \begin{align*}\[simplicial space\end{align*} ] \(W \Delta^{\operatorname{op}}\to {\mathsf{sSet}}_{{\mathsf{Kan}}}\) with some conditions.
\begin{align*}\[simplicial set\end{align*} ] \({\mathsf{sSet}}^{\Delta^{\operatorname{op}}}\) has a model structure where complete Segal spaces are the fibrant objects
Exercise: if \(W\) is a complete Segal space then \(i_1^* W\) is a quasicategory where \(i_1: \Delta \to \Delta^{\times 2}\) sends \([n]\) to \(([n], [0])\).
Limits: isomorphisms on hom sets, or terminal objects in the \begin{align*}\[cone category\end{align*} ] :
\begin{align*}\[attachments/image_2021-06-05-12-24-46.png\end{align*} ]
Terminal objects: isomorphisms from \begin{align*}\[slice category\end{align*} ] to \(\mathsf{C}\):
\begin{align*}\[attachments/image_2021-06-05-12-25-36.png\end{align*} ]
Enriched definition of limits:
\begin{align*}\[attachments/image_2021-06-05-12-27-23.png\end{align*} ]
Terminal objects: \(X_{/ {x}} \xrightarrow{\sim} X\) is a \begin{align*}\[weak equivalence\end{align*} ] in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\).
All definitions of limits in \({ \underset{(\infty, {1})}{ \mathsf{Cat}} }\) recover the usual notions via the nerve.
Exercise: Show that the limit of \(F: I\to \mathsf{C}\) is the limit of its \begin{align*}\[nerve\end{align*} ]?
Theorem: limit of \(F\) in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\) is a \begin{align*}\[homotopy limit\end{align*} ] of its \begin{align*}\[adjoint\end{align*} ] in \({\mathsf{sSet}}{\hbox{-}}\mathsf{Cat}_{{\mathsf{Kan}}}\), and the limit of its adjoint in \({\mathsf{sSet}}^{\Delta^{\operatorname{op}}}_{ \mathsf{CSS} }\).
Upshot: many different models, can move between different models.
Things to look up from written notes:
Next: models of \({ \underset{\infty}{ \mathsf{Cat}} }{\infty, 2}\)
\begin{align*}\[base change along a functor\end{align*} ]??
\begin{align*}\[2-category\end{align*} ] : categories \begin{align*}\[enriched\end{align*} ] in categories
Recall: \({ \mathsf{CSS} } = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{sSet}})\).
Idea: keep track of 2-morphisms, i.e. two-cells, can keep all of their possible compositions
\begin{align*}\[attachments/image_2021-06-05-13-09-28.png\end{align*} ]
\begin{align*}\[attachments/image_2021-06-05-13-09-54.png\end{align*} ]
\begin{align*}\[attachments/image_2021-06-05-13-10-05.png\end{align*} ]
Problem: for many \begin{align*}\[QFT\|QFTs\end{align*} ], we don’t know how to write down the quantum \begin{align*}\[observables\end{align*} ] \({\mathcal{F}}(U)\) for an open \(U \subseteq X\) (e.g. for \(X\) spacetime).
Three approaches:
Factorizable cosheaves (topological/differential geometric) Quantum observables in the field theory.
Vertex algebras (algebra and analysis) Infinite dimensional vector spaces, symmetries of \(2d\) conformal field theories
Chiral or \begin{align*}\[factorization algebra\|factorization algebras\end{align*} ] (algebraic geometry)
\begin{align*}\[quasicoherent sheaf\|quasicoherent sheaves\end{align*} ] (so \begin{align*}\[D modules\end{align*} ]) with \begin{align*}\[Lie algebra\|Lie (co)algebra\end{align*} ] structures. Collisions between local operators
\begin{align*}\[vertex algebra\end{align*} ], meromorphic multiplication \(V^{\otimes 2} \to V((z))\).
\begin{align*}\[vertex operator\end{align*} ] : \(Y({-}, z): V\to \mathop{\mathrm{End}}V { \left[ {z, z^{-1}} \right] }\) where \(A\mapsto \sum A_{(n)} z^?\) where \(A_{(n)}:V\to V\) should be thought of as ways of multiplying.
Any commutative algebra with a \begin{align*}\[derivation\end{align*} ] \(T\) yields a vertex algebra \(Y(A, z) = e^{zT} A = \sum _{T^k A \over k!} z^k\).
The \begin{align*}\[monster group\end{align*} ] is the largest sporadic simple group, constructed as the automorphisms of a \begin{align*}\[vertex algebra\end{align*} ] constructed from the \begin{align*}\[Leech lattice\end{align*} ].
We knew the dimensions of representations before the construction (e.g character tables), conjectured to be related to \begin{align*}\[modular functions\end{align*} ], Borcherds Fields in 98 for proving this!
Important fact: certain categories of representations of affine Lie algebras/quantum groups form \begin{align*}\[modular tensor categories\end{align*} ] : Kazhdan-Lusztig 93!
Nice invertible objects? Levels are closed under tensor?
Beilinson-Drinfeld, 90s: factorization/chiral algebras
Discrete example: particles on a surface labeled with integers, where colliding causes addition of labels.
Ex: the \begin{align*}\[Hilbert scheme\end{align*} ] of points. Lengths of subschemes equals dimension of quotient of \(\operatorname{Spec}\) as a vector space over \({\mathbb{C}}\). Consider \(\operatorname{Spec}{\mathbb{C}}[x, y] / \left\langle{ x, y(y- \lambda) }\right\rangle\). \(\lambda=0\) remembers that the collision happened along the \(y\) axis.
\(\operatorname{Hilb}_X\) is smooth when \(\dim X = 1,2\).
Most important example of a factorization space: \begin{align*}\[Beilinson-Drinfeld Grassmannian\end{align*} ].
Upshot: combine all 3 approaches to tackle problems!
\begin{align*}\[vertex algebra\end{align*} ] : a factorization algebra over curves (with more symmetry)
A vertex algebra is quasi-conformal if it has a nice action of \(\mathop{\mathrm{Aut}}\operatorname{Spf}{\mathbb{C}} { \left[ {t} \right] }\), automorphisms of a formal disk? See \begin{align*}\[formal spectrum\end{align*} ].
(Corresponds to Virasoro symmetry of the CFT).
Note: aut of formal disk is more like in ind object in \begin{align*}\[group scheme\|Group schemes\end{align*} ]?
Not an \begin{align*}\[algebraic group\end{align*} ], carries some limits/colimits?
Direct bridge from \begin{align*}\[factorizable cosheaves\end{align*} ] to factorizable algebras doesn’t quite exist yet!
Reference: eAKTs
Tags: #seminar_notes #k_theory
Refs:
\begin{align*}\[Brauer group\end{align*}
]
\begin{align*}\[Azumaya algebra\end{align*}
]
This is some \(H^2\) perhaps? Like \(\mathop{\mathrm{Br}}(X) = H^2(X; {\mathbb{G}}_m)\)? Need to figure out what kind of cohomology this is though.
See \begin{align*}\[Brauer-Manin pairing\end{align*} ], \begin{align*}\[Tate pairing\end{align*} ]
degree of cycles on \begin{align*}\[Chow ring\|Chow\end{align*} ]
See \begin{align*}\[central fiber\end{align*} ], \begin{align*}\[formal scheme\end{align*} ]
\(\lim^1\), see \begin{align*}\[lim1\end{align*} ]
See \begin{align*}\[GAGA\end{align*} ]
Morita theory: for \(R\in \mathsf{Ring}, A,B\in {\mathsf{Alg}}_{R}\), \(A\sim B\) are \begin{align*}\[Morita equivalent\end{align*} ] iff \({\mathsf{A}{\hbox{-}}\mathsf{Mod}} \equiv \mathsf{B}{\hbox{-}}\mathsf{Mod}\), and \(A\) is \begin{align*}\[Azumaya algebra\|Azumaya\end{align*} ] if it’s \begin{align*}\[invertible object of a category\end{align*} ] in the following sense: there is an \(A'\) such that \(A\otimes A' \sim R\)
What is \begin{align*}\[presentable infinity category\end{align*} ]?
Part of an equivalence: take a \begin{align*}\[compact generator\end{align*} ], take its endomorphism algebra, take category of modules over that algebra?
See \begin{align*}\[etale descent\end{align*} ] and \begin{align*}\[Zariski descent\end{align*} ].
\begin{align*}\[invertible object of a category\end{align*} ] implies \begin{align*}\[dualizable object of a category\end{align*} ] but not conversely.
Smooth and proper: dualizable?
See \begin{align*}\[perfect complexes\|perfect complex\end{align*} ]
Reference: ???, GROOT
Big idea: free implies flat for algebras, is this true in the equivariant settings?
Abelian groups \(\approx\) \begin{align*}\[Mackey functor\end{align*} ].
\({\mathbb{Z}}\approx\) Burnside Mackey functors
Commutative rings \(\approx\) \begin{align*}\[Green functor\end{align*} ] \((E_\infty\) algebras), Incomplete functors, \begin{align*}\[Tambara functor\end{align*} ]
Free algebra \({\mathbb{Z}}[G]\) comparable to free incomplete Tambara functor
Similarities come from being algebras over \begin{align*}\[Operads\end{align*} ].
\begin{align*}\[Hill-Hopkins-Ravenel\end{align*} ] involves spectral sequences of Mackey functors
All rational Mackey functors are free
\(A^{\mathcal{O}}[x_{G/H}]\) is almost never flat.
A Mackey functor is an additive functor \(M: A^g\to {\mathsf{Ab}}\), where \(A^G\) is the \begin{align*}\[Burnside category\end{align*} ] : finite \(G{\hbox{-}}\)sets, where morphisms \(A^G(X, Y)\) is the group completion wrt \(\coprod\) of finite \(G{\hbox{-}}\)sets, so \begin{align*}\[spans\end{align*} ].
To define a Mackey functor \(F\), it suffices to give abelian groups \(F(G/H)\) for \(H\leq G\), restrictions \(\operatorname{res}^H_K\), and \begin{align*}\[transfer map\end{align*} ] \({\mathrm{tr}}_K^H\) in the target.
\begin{align*}\[Burnside Mackey functor\end{align*} ] : \(\underline{A}\).
Theorem (Lewis): the category of Mackey functors is abelian, and has a \begin{align*}\[symmetric monoidal category\|symmetric monoidal\end{align*} ] product \(\boxtimes\) with unit \(\underline{A}\).
A \begin{align*}\[Green functor\end{align*} ] is a monoid for \(\boxtimes\), which is an \begin{align*}\[E_n ring spectrum\|E_infty algebra\end{align*} ] in Mackey functors.
An incomplete Tambara functor is an \(N_\infty\) algebra in Mackey functors
A \begin{align*}\[Tambara functor\end{align*} ] is a \begin{align*}\[Green functor\end{align*} ] with that data of a \begin{align*}\[norm\end{align*} ] map \(\nm_K^H\), a multiplicative morphism.
Indexing systems: valid suborderings on the poset lattice of subgroups
Theorem (Barnes-Roitzheim-?) For \(C_{pq}\), there are roughly a \begin{align*}\[Catalan's number\end{align*} ] of valid indexing systems.
Tags: #idle_thoughts
Really cool idea I like from that talk: what is the probability density of objects in a category?
In a precise sense, what proportion of objects are projective, flat, free, dualizable, indecomposable, simple, etc?
I think I really want analytic structure on a category! I’m reminded of results like \begin{align*}\[Morse functions\end{align*} ] being generic in spaces of functions, or perturbing \begin{align*}\[Hamiltonian\|Hamiltonians\end{align*} ] in \begin{align*}\[Floer Theory\end{align*} ]. We can cook up topologies to make these kinds of statements precise in the classical setting….how can we do it here?
Look up \begin{align*}\[simple normal crossings\end{align*} ] divisor.
The \begin{align*}\[Gelfand representation\end{align*} ] is really cool. Look into how this duality shows up for schemes!
What is the \begin{align*}\[length of a module\end{align*} ]?
Tags: #idle_thoughts #category_theory #infinity_cats
Terrible attempt at a way around ZFC: axiomatically define a category the way one axiomatizes Euclidean geometry:
This makes the definition infinitely recursive, which might be a problem. One could truncate this by asking the 1st iteration of taking “the hom category” to result in a discrete category: some objects but no morphisms between distinct objects. This is definitely taken care of by \begin{align*}\[infinity categories\end{align*} ].
Define the category of sets by specifying a single point as an initial object, then freely taking powersets and unions. I think you at least get something whose nerve is the same as the nerve of the category of finite sets. I think one can also realize these operations at the categorical level: powersets are like exponentials \(2^X\), you can get disjoint unions from limits, and maybe usual unions/intersections from pushouts/pullbacks?
Tags: #seminar_notes #geometric_topology #floer Refs: \begin{align*}\[Heegard-Floer homology\end{align*} ]
Reference: Kristen Hendricks, Surgery formulas for involutive Heegaard Floer homology. Stanford Topology Seminar.
Want to study homology \begin{align*}\[cobordism groups\end{align*} ] of \begin{align*}\[3-manifolds\end{align*} ] \(\Theta_{\mathbb{Z}}^3\).
See \begin{align*}\[Seifert fibered spaces\end{align*} ].
People like \begin{align*}\[HF\end{align*} ] because there are a lot of computational tools! In particular, a \begin{align*}\[surgery formula\end{align*} ].
Check out \begin{align*}\[ideal sheaves\end{align*} ], \begin{align*}\[Birdgeland stability conditions\end{align*} ].
“Rotate” an exact sequence to get an \begin{align*}\[exact triangle\end{align*} ] :
What is an \begin{align*}\[etale algebra\end{align*} ]?
See \begin{align*}\[Serre's uniformity conjecture\end{align*} ].
Source: Frobenius exact symmetric tensor categories - Pavel Etingof. IAS Geometric/modular representation theory seminar. https://www.youtube.com/watch?v=7L06K7SL5qw
Tags: #seminar_notes #representation_theory #category_theory #monoidal Refs: \begin{align*}\[tensor category\end{align*} ]
Looking at \begin{align*}\[modular representations\end{align*} ] of finite groups.
See \begin{align*}\[tensor ideal\end{align*} ], \begin{align*}\[Krull-Schmidt theorem\end{align*} ] : decomposition into indecomposables is essentially unique.
Can take \begin{align*}\[split Grothendieck ring\end{align*} ].
\begin{align*}\[Symmetric tensor category\end{align*} ] \(\mathsf{C}\):
Example: \({\mathsf{Rep}}_k(G)\) the category of finite-dimensional representations of \(G\). Take \(G=1\) to recover \({ \mathsf{Vect} }_{/k}\).
Such a category is \begin{align*}\[tannakian\end{align*} ] if there exists a \begin{align*}\[fiber functor\end{align*} ] : a symmetric tensor functor \(F: \mathsf{C} \to { \mathsf{Vect} }_{/k}\).
For an ( \begin{align*}\[additive category\end{align*} ] rigid symmetric monoidal category \(\mathsf{C}_{/k}\) with \(\mathop{\mathrm{End}}_{\mathsf{C}}(\one) = k\), for any \(f\in \mathsf{C}(X, X)\) we can define its \begin{align*}\[trace (monoidal categories)\|trace\end{align*} ] \(\operatorname{Tr}(f) \in \mathop{\mathrm{End}}_{\mathsf{C}}(\one)\):
What is an \begin{align*}\[isocrystal\end{align*} ]?
What does it mean to be \begin{align*}\[crystalline\end{align*} ]?
What is a \begin{align*}\[symplectic basis\end{align*} ]?
What is the \begin{align*}\[Mordell-Weil sieve\end{align*} ]?
How can one pass from p-adic solutions to rational solutions?
What is the \begin{align*}\[conductor of an elliptic curve\end{align*} ]?
What is \begin{align*}\[Serre's open image theorem\end{align*} ].
What is \begin{align*}\[inertia\end{align*} ]?
What is the \begin{align*}\[cyclotomic character\end{align*} ]?
See \begin{align*}\[Mazur Program B\end{align*} ]
See \begin{align*}\[Goursat's lemma in group theory\end{align*} ].
We haven’t been able to classify the rational points on \begin{align*}\[modular curves\end{align*} ]!
Reference: Kirsten Wickelgren, Colloquium Presentation: zeta functions and a quadratic enrichment. Rational Points and Galois Representations workshop
Tags: #seminar_notes #homotopy_theory #algebraic_geometry #stable_homotopy #motivic Refs: \begin{align*}\[motivic homotopy\end{align*} ]
See \begin{align*}\[Atiyah duality\end{align*} ] : define the dual of \(M\) as \(M^{-{\mathbf{T}}M}\), the \begin{align*}\[Thom space\|Thom space\end{align*} ] of (minus) the tangent bundle.
Define the \begin{align*}\[trace (monoidal categories)\|trace\end{align*} ] :
Then \(\operatorname{Tr}(\phi) \in \mathop{\mathrm{End}}_{\mathsf{C}}(\one, \one)\) is an endomorphism of the unit.
Example: \begin{align*}\[Lefschetz fixed point theorem\end{align*} ], \begin{align*} \operatorname{Tr}(\phi) = \sum_{x\in M, \phi(x) = x} \operatorname{Ind}_x \phi \in \mathop{\mathrm{End}}_{{\mathsf{ho}}{\mathsf{Sp}}}(\one) \xrightarrow{\deg \,\, \sim} {\mathbb{Z}} ,\end{align*} where we take the degree of a map between spheres.
\({\mathbb{S}}= \one \in {\mathsf{ho}}{\mathsf{Sp}}\).
Use that \(H^*({-}, {\mathbb{Q}})\) preserves tensor products, and apply the Kunneth formula: \begin{align*} H^*(\operatorname{Tr}(\phi)) &= \operatorname{Tr}(H^*(\phi)) \\ \implies \sum (-1)^i \operatorname{Tr}( H^i(\phi); H^i(M) {\circlearrowleft}) &= \sum_{x\in M, \phi(x) = x} \operatorname{Ind}_x \varphi .\end{align*}
Rationality of \(\zeta\):
Use hocolims to glue spaces, but may not work in schemes.
Example: take \(X \mathrel{\vcenter{:}}={\mathbb{P}}^n/{\mathbb{P}}^{n-1}\), then we’d want \(X({\mathbb{C}}) \cong S^{2n}\) and \(X({\mathbb{R}}) \cong S^n\)
Problem: this quotient isn’t a \begin{align*}\[scheme\end{align*} ]. Can freely add these limits.
We want \({\mathbb{P}}^i / {\mathbb{P}}^{i-1}\) to be the building blocks or cells
Morel and Voevodsky, \({\mathbb{A}}^1\) \begin{align*}\[stable homotopy category\end{align*} ] over \(k\), denoted \({\mathsf{SH}}(k)\).
Take an analog of degree, the Morel degree: \begin{align*} \deg: [{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1} ] \xrightarrow{} {\operatorname{GW}}(k) .\end{align*}
\begin{align*}\[Grothendieck-Witt\end{align*} ] group: formal differences of isomorphism classes of nondegenerate symmetric \begin{align*}\[bilinear forms\end{align*} ].
Special form: the hyperbolic form \begin{align*} \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle = \left\langle{a}\right\rangle + \left\langle{-a}\right\rangle = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} .\end{align*}
See \begin{align*}\[rank\end{align*} ], \begin{align*}\[signature\end{align*} ], \begin{align*}\[discriminant\end{align*} ] of \begin{align*}\[quadratic form\|quadratic forms\end{align*} ].
Trace here will take values in \({\operatorname{GW}}(k)\).
\begin{align*}\[Lefschetz fixed point theorem\end{align*} ] due to Hoyois:
Notation: \(dZ^{{\mathbb{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log \zeta^{{\mathbb{A}}^1}(t) = \sum_{m\geq 1} \operatorname{Tr}(\phi^m)t^{m-1}\).
Prop: \begin{align*} \operatorname{rank}dZ^{{\mathbb{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log .\end{align*}
See Kapranov \begin{align*}\[motivic zeta function\end{align*} ] :
See \begin{align*}\[Euler class\end{align*} ], \begin{align*}\[Hopf map\end{align*} ]
Major point: this is genuinely something new, isn’t just recovered by taking the compactly supported euler characteristic:
Tags: #THH #factorization_homology #seminar_notes Refs: \begin{align*}\[nonabelian Poincare duality\end{align*} ], \begin{align*}\[factorization homology\end{align*} ]
Reference: Foling Zou, Nonabelian Poincare duality theorem and equivariant factorization homology of Thom spectra. MIT Topology Seminar.
Goal: want to formulate \begin{align*}\[monads\end{align*} ] and \begin{align*}\[operads\end{align*} ] categorically.
See \begin{align*}\[lambda sequences\end{align*} ], something like a functor \({\mathsf{FinSet}}^{\operatorname{op}}\to \mathsf{C}\)?
See \begin{align*}\[Day convolution\end{align*} ] as an example of a monoidal product.
See \begin{align*}\[monadic bar construction\end{align*} ] and monoidal \begin{align*}\[bar construction\end{align*} ].
Examples of \begin{align*}\[factorization homology\end{align*} ] : \begin{align*} \int_{S^1}A &&\simeq{\operatorname{THH}}(A) \\ \int_{T^n}A &&\simeq{\operatorname{THH}}^n(A) && \text{iterated THH} .\end{align*}
For \(\sigma\) the \begin{align*}\[sign representation\end{align*} ], \(\int_{S^\sigma} A \simeq\operatorname{THR}(A)\) for \(E_\sigma{\hbox{-}}C_2\) spectra.
See Horev, Hessolholt-Madsen.
Axiomatic approach to factorization homology: take a left \begin{align*}\[Kan extension\end{align*} ] of the following:
Can compute \begin{align*}\[Kan extension\end{align*} ] via the \begin{align*}\[bar construction\end{align*} ].
Theorem: \begin{align*}\[equivariant\end{align*} ] \begin{align*}\[nonabelian Poincare duality\end{align*} ] :
What is the \begin{align*}\[virtual dimension\end{align*} ] of a bundle?
\({\operatorname{Pic}}\): subcategory of invertible objects, \begin{align*}\[Picard group\|PIcard groupoid\end{align*} ]
\begin{align*}\[Thom spectrum\end{align*} ] functor:
where \(R{\hbox{-}}\)line is the \(\infty{\hbox{-}}\)category of line bundles up to equivalence?
Preserves \(G{\hbox{-}}\)colimits, so formally the Thom spectrum functor commutes with factorization homology.
In proof of theorem, use \begin{align*}\[nonabelian Poincare duality\end{align*} ] to reduce a complicated gadget to a mapping space.
For \(\operatorname{THR}\) on the algebra side, see Teena Gerhardt’s work? Haynes Miller suggests looking at the \begin{align*}\[de Rham-Witt complex\end{align*} ]?
Not every \begin{align*}\[simplicial complex\end{align*} ] is a \begin{align*}\[PL\end{align*} ] manifold:
\begin{align*}\[Arpon Raksit - Hochschild homology and the derived de Rham complex revisited\end{align*} ]
\begin{align*}\[Andrew Blumberg, Floer homotopy theory and Morava K-theory\end{align*} ]
\begin{align*}\[Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups\end{align*} ]
^22ba3a
Reference: Padmavathi Srinivasan, UGA NT Seminar.
If \(r\) is the rank and \(g\) is the genus for \(X\) a nice curve over \({\mathbb{Q}}\) with good reduction at \(p\),
See p-adic \begin{align*}\[height pairing\end{align*} ].
Can compute heights as intersection numbers on regular models.
See \begin{align*}\[Abel-Jacobi map\end{align*} ]
Try to realize rational points as the zero locus of p-adic analytic functions.
Recent work: quadratic Chabauty used to find all rational points on the infamously cursed \begin{align*}\[modular curve\end{align*} ] \(X_S(13)\).
Looking for explanations through \begin{align*}\[Arakelov theory\end{align*} ] instead of \begin{align*}\[p-adic Hodge theory\end{align*} ].
See \begin{align*}\[Mordell-Weil rank\end{align*} ]
See \begin{align*}\[Néron-Severi\end{align*} ] of the \begin{align*}\[Jacobian\end{align*} ].
Get height functions where local heights \(h_p\) can be computed by iterated \begin{align*}\[Coleman integrals\end{align*} ].
There is a canonical height machine for abelian varieties.
Symmetric line bundles: \(\mathcal{L}\cong [1]^* \mathcal{L}\).
Zhang defines a \begin{align*}\[metric on a line bundle\end{align*} ], which gives a way to measure the size of elements in each fiber.
When you have an integral model: take closures!
See \begin{align*}\[valuations\end{align*} ], used to define admissible metrics.
Admissible metrics on \({\mathcal{O}}_X\) factor through the \begin{align*}\[reduction graph\end{align*} ]
Adelic metric: a collection of metrics for almost every \begin{align*}\[place\end{align*} ].
Rigidified bundles: remember a point in the fiber.
For each place, there is a canonical metric which makes certain isomorphisms into isometries in a \begin{align*}\[Banach space\end{align*} ]
Curvature form \(\mathop{\mathrm{Curv}}(\mathcal{L}_v)\) is sent to the first \begin{align*}\[Chern class\end{align*} ] \(c_1(\mathcal{L}_v )\) under cup product in de Rham cohomology.
To get a canonical metric for all line bundles, it suffices to canonically metrize the \begin{align*}\[Poincare bundle\end{align*} ]. Every line bundle on \(A\) is a pullback of it.
Any two metrics with the same curvature differ by \(\int \omega\) for \(\omega\).
Reference: The reciprocity law for the twisted second moment of Dirichlet L-functions https://arxiv.org/pdf/0708.2928.pdf
What is a \begin{align*}\[Dirichlet character\end{align*} ]?
What is a \begin{align*}\[Gauss sum\end{align*} ]?
What is the completion of an \begin{align*}\[L function\end{align*} ]? Guessing this has to do with continuation.
What is Dirichlet’s trick?
How can you break a sum up into \begin{align*}\[arithmetic progressions\end{align*} ]?
Reference: The \(K'\)-theory of \begin{align*}\[monoid\end{align*} ] sets https://arxiv.org/pdf/1909.00297.pdf
\begin{align*}\[K-Theory\end{align*} ]
The category \({\mathsf{FinSet}}_{{\scriptstyle { * } }}\) (see \begin{align*}\[Finset\end{align*} ] ) of finite pointed sets is quasi-exact, and \begin{align*}\[Barratt-Priddy-Quillen\end{align*} ] implies that \(K({\mathsf{FinSet}}_{\scriptstyle { * } }) \simeq{\mathbb{S}}\).
\begin{align*}\[Group completion\end{align*} ] : comes from \({\Omega}^\infty {\Sigma}^\infty {\mathbf{B}}G_+\).
Big theorem: \begin{align*}\[Devissage\end{align*} ]. But I have no clue what this means. Seems to say when \({\mathsf{K}}(A) \cong {\mathsf{K}}(B)\)?
Apparently easy theorem: \({\mathsf{K}}'({\mathbb{N}}) \simeq{\mathbb{S}}\).
The \begin{align*}\[Picard group\end{align*} ] of \({\mathbb{P}}^1\) shows up:
Tags: #seminar_notes #algebraic_geometry
Reference: Stefan Schreieder, Refined unramified cohomology. Harvard/MIT AG Seminar talk.
See the \begin{align*}\[Chow ring\end{align*} ] and \begin{align*}\[cycle class map\end{align*} ]. Understanding the image amounts to the \begin{align*}\[Hodge conjecture\end{align*} ] and understanding torsion in the image \(Z^i(X)\)?
\begin{align*}\[Gysin sequence\end{align*} ] yields a residue map \({\partial}_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)\).
Interesting parts of the Coniveau spectral sequence: something coming from unramified cohomology, and something coming from algebraic cycles mod algebraic equivalence.
Failure of integral \begin{align*}\[Hodge conjecture\end{align*} ] :
Uses \begin{align*}\[Bloch-Kato conjecture\end{align*} ]
Allows detecting classes in \(Z^2(X)\) using \begin{align*}\[K-Theory\end{align*} ] methods.
See \begin{align*}\[Borel-Moore cohomology\end{align*} ] – for \(X\) a smooth \begin{align*}\[algebraic scheme\end{align*} ], essentially singular homology with a degree shift?
See \begin{align*}\[pro-objects\end{align*} ] and \begin{align*}\[ind-objects\end{align*} ] in an arbitrary category.
Filter by codimension, then obstructions to extending over higher codimension things is measured by cohomology of the \begin{align*}\[Function field\end{align*} ] :
Here \({{\partial}}\) is a residue map.
See \begin{align*}\[separated\end{align*} ] schemes of \begin{align*}\[finite type\end{align*} ].
Main theorem, works not just for smooth schemes, but in greater generality:
Torsion in the Griffiths group is generally not finitely generated.
See \begin{align*}\[canonical class\end{align*} ] \(K_S\) for a surface, \begin{align*}\[Abel-Jacobi invariants\end{align*} ]?
No Poincaré duality for Chow groups, at least not at the level of cycles. Need to pass to cohomology.
Theorem: there exists a \begin{align*}\[regular\end{align*} ] \begin{align*}\[flat morphism\end{align*} ] \begin{align*}\[proper\end{align*} ] \(S\to \operatorname{Spec}{\mathbb{C}}{\left[\left[ t \right]\right] }\) such that \(S_\eta\) is an Enriques surface, \(S_0\) is a union of \begin{align*}\[ruled surfaces\end{align*} ], and \(\mathop{\mathrm{Br}}(S) \twoheadrightarrow\mathop{\mathrm{Br}}(S_\eta)\).
See \begin{align*}\[Zariski locally\end{align*} ] and \begin{align*}\[étale locally\end{align*} ].
\begin{align*}\[unramified cohomology\end{align*} ] is linked to \begin{align*}\[Milnor K theory\end{align*} ].
Reference: https://www.youtube.com/watch?v=XTOwj1LvntM
Clausen: a baby topic in \begin{align*}\[geometric representation theory\end{align*} ] is \begin{align*}\[Springer correspondence\end{align*} ].
Tags: #representation_theory
See \begin{align*}\[Hopf algebra\end{align*} ]
Why \begin{align*}\[Hopf algebra\end{align*} ]? Some natural examples:
\(kG\) the \begin{align*}\[group algebra\end{align*} ].
\(k^G = \mathop{\mathrm{Hom}}_k(kG, k)\) an algebra of functions, forcing distinct group elements to be orthogonal idempotents, take \(\left\{{ P_x {~\mathrel{\Big\vert}~}x\in G }\right\}\) with \(P_x P_y = \delta_{xy} P_y\) ??
Consider category \(\mathsf{H}{\hbox{-}}\mathsf{Mod}^{\mathrm{fd}}\) of finite-dimensional \begin{align*}\[Representation theory\end{align*} ] of \(H\).
Antipode will be invertible when \(H\) is finite dimensional
A lot of structures here: closed under tensors, duals, contains \(k\).
Finite \begin{align*}\[tensor category\end{align*} ] : looks like \(\mathsf{H}{\hbox{-}}\mathsf{Mod}\), \begin{align*}\[Enriched category\end{align*} ] over vector spaces, \begin{align*}\[Monoidal category\end{align*} ], coherent associativity via \begin{align*}\[pentagon axiom\end{align*} ], \begin{align*}\[triangle axiom\end{align*} ].
Define \(\operatorname{Ext} ^n_{\mathsf{C}}(X, Y)\) to be equivalence classes of \(n{\hbox{-}}\)fold extensions, i.e. exact sequences \(0 \to Y \to E_n \to \cdots \to E_1 \to X \to 0\), and \(H^*(\mathsf{C}) \mathrel{\vcenter{:}}= H^*_{\mathsf{C}}(\one, \one) = \bigoplus _{n\geq 0} \operatorname{Ext} ^n_{\mathsf{C}} (\one, \one )\). Can similarly replace \(\one\) with \(X\) to define \(H^*(X)\), which will be a module over \(H^*(\mathsf{C})\).
\begin{align*}\[support variety\end{align*} ] : \(V_{\mathsf{C}}(\one) = \operatorname{mSpec}H^*(\mathsf{C})\), \(V_{\mathsf{C}}(X)\) is a more complicated quotient.
Representation theory of categories: module categories over a category!
Big question: tensor product property. Is there an equality \begin{align*} V_{\mathsf{C}}(X\otimes Y) \overset{?}{=} V_{\mathsf{C}}(X) \cap V_{\mathsf{C}}(Y) .\end{align*}
True for cocommutative \begin{align*}\[Hopf algebra\end{align*} ], some \begin{align*}\[quantum groups\end{align*} ].
Some counterexamples in non-braided monoidal categories. Uses a smash product of modules
See \begin{align*}\[thick ideals\end{align*} ].
Tags: #k_theory #adic #seminar_notes
Reference: Clausen, the K-theory of adic spaces. https://www.youtube.com/watch?v=e_0PTVzViRQ
\begin{align*}\[adic spaces\end{align*} ] : formalism for non-Archimedean geometry.
\begin{align*}\[Formal scheme\end{align*} ] : e.g. formal thickening of a subvariety. Sometimes want to delete a \begin{align*}\[special fiber\end{align*} ].
Definition of ( \begin{align*}\[adic ring\end{align*} ], complete with respect to a finitely-generated ideal, so \(R = \varprojlim R/I^n\)
Some nice features:
Subtleties:
\(\mathsf{Solid}_{\mathbb{Z}}\): abelian bicomplete category of \begin{align*}\[solid mathematics\|solid sets\end{align*} ] Full subcategory of \begin{align*}\[condensed sets\end{align*} ]. Has compact projective generators \(\prod_I {\mathbb{Z}}\)
For morphisms, note \(\mathop{\mathrm{Hom}}( \prod_I {\mathbb{Z}}, {\mathbb{Z}}) = \bigoplus_I {\mathbb{Z}}\)
Let \(\mathsf{Perf}\) be \begin{align*}\[perfect complexes\end{align*} ], why not consider \(K({ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }(X))\) or \(K(\mathsf{Perf}(X))\)?
Need a good category of \begin{align*}\[quasicoherent sheaves\end{align*} ]
What is a \begin{align*}\[presentable category\end{align*} ]?
What is a \begin{align*}\[Tate algebra\end{align*} ]?
What is \begin{align*}\[Arakelov theory\end{align*} ]?
\begin{align*}\[Arithmetic Statistics\end{align*} ] {#notes-on-arithmetic-statistics}
Interesting question in arithmetic statistics: for \(G \in {\mathsf{Grp}}\) finite, how many Galois extensions are there \(K/{\mathbb{Q}}\) with \(G = { \mathsf{Gal}} (K/{\mathbb{Q}})\) and \(\Delta \leq N\) ( \begin{align*}\[discriminant\end{align*} ] for some fixed \(N\)?
Example
One can ask a similar question about \({ \operatorname{Cl}} (K)\) for \(G\in {\mathsf{Ab}}\), or replacing \({\mathbb{Q}}\) with \begin{align*}\[function field\end{align*} ] \({\mathbb{F}}_q(t)\) for \(q=p^n\), and ask questions about frequency of primes \begin{align*}\[ramified primes\end{align*} ], \begin{align*}\[split primes\end{align*} ], or remaining \begin{align*}\[inert primes\end{align*} ].
Cool fact: there is an \begin{align*}\[equivalence of categories\end{align*} ] between finitely-generated extensions \(K/k\) with \(\operatorname{trdeg}(K/k) = 1\) and regular projective \begin{align*}\[curves\end{align*} ] \(C_{/k}\).
\begin{align*}\[Hurwitz spaces\end{align*} ] come up here!
\({\varepsilon}\) is a \(q^i\) \begin{align*}\[Weil number\end{align*} ] if \({\left\lvert { \iota({\varepsilon}) } \right\rvert} = q^{i/2}\) for any embedding \(\iota: { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\hookrightarrow{\mathbb{C}}\).
As a general philosophy, one should expect that \begin{align*}\[moduli space\end{align*} ] problems whose objects have nontrivial automorphisms are representable by \begin{align*}\[stack\end{align*} ], and those without nontrivial automorphisms are \begin{align*}\[representable\end{align*} ] by \begin{align*}\[scheme\end{align*} ].
\begin{align*}\[Weil Conjectures Talks\end{align*} ]
Some old notes from March 10th, 2020
I talked to Erik Schreyer today about some of the research he did with his advisor Jason Cantarella, including his dissertation work (which he spoke about in the Geometry seminar last week) and a few other papers.
His dissertation work involved a cool way to represent arbitrary planar curves by piecewise circular arcs:
From what I understand, this involves fixing a curve (blue), choosing a collection of circles \(C_1, \cdots C_n\) (black) such that each \(C_i\) intersects \(C_{i+1}\) in at least one distinguished point \(p_i\) (pink). The curve traced out by following an arc on \(C_i\) and switching to circle \(C_{i+1}\) at \(p_i\) is intended to yield a good approximation to the original curve, with certain regularity conditions at the \(p_i\) (such as the first derivatives along both arcs agreeing at the point).
Erik’s work actually seems to go a bit farther – he has an algorithm (a curve-closing operator) that actually takes an open curve and produces a closed curve that is nearby in the \(C_1\) norm. He uses this to construct piecewise circular approximations that consist of circles of equal radii, along with some control over the \(C^1\) distance between the original curve and the approximation.
We also talked a bit about another problem Jason was working on, discussed in the following papers:
The symplectic geometry of closed equilateral random walks in 3-space (Cantarella, Shonkwiler 2013)
A Fast Direct Sampling Algorithm for Equilateral Closed Polygons (Cantarella et al 2015)
Dirichlet’s Theorem: An arithmetic progress with \((a, p) = 1\) contains infinitely many primes. As a corollary, one can always find a prime \(q\) that generates \({\mathbb{Z}}_p^{\times}\) for any prime \(p\).
Not every sequence of the form \(0\to A \to A \oplus C \to C \to 0\) splits; take \begin{align*} 0 \to {\mathbb{Z}}\to {\mathbb{Z}}\oplus \bigoplus_{\mathbb{N}}{\mathbb{Z}}/(2) \to \bigoplus_{\mathbb{N}}{\mathbb{Z}}/(2) \to 0 \end{align*} where the first map is multiplication by 2, the second is the quotient map and a right-shift. This can’t split because \((1, 0, \cdots)\) has order 2 in the RHS but pulls back to \((1, 0) \oplus (2{\mathbb{Z}}\oplus 0)\) which has no element of order 2.
See \begin{align*}\[cogroup\end{align*} ].
Reference: Remy van Dobben de Bruyn (Princeton and IAS), “Constructing varieties with prescribed Hodge numbers modulo m in positive characteristic.” Stanford AG Seminar.
One of Serre’s tricks: use group quotients cleverly.
There is a way to directly add/subtract/multiply \begin{align*}\[Hodge diamonds\end{align*} ].
Inverse Hodge problem: when can a Hodge diamond be realized by a smooth \begin{align*}\[projective variety\end{align*} ]? Very hard problem. Want to use this to get information about \(h_{{\mathrm{crys}}}\), i.e. \begin{align*}\[Crystalline cohomology\end{align*} ].
Easier question: look at linear/polynomial relations satisfied by all Hodge diamonds of \({\mathsf{Var}_{/k} }({\mathsf{sm}}, \mathop{\mathrm{proj}})\)?
Main theorems: for a fixed dimension \(n\),
Important tools:
\begin{align*} h(\operatorname{Bl}_2 X) &= h(X) - h(z) + h(E) \\ &= h(X) - h(z) + h(z)(1 + {\mathbb{L}}+ {\mathbb{L}}^2 + \cdots + {\mathbb{L}}^{c-1} )\\ &= h(z) + ({\mathbb{L}}+ {\mathbb{L}}^2 + \cdots + {\mathbb{L}}^{c-1} ) ,\end{align*}
where \(z\) is the point removed and \(E\) is the \begin{align*}\[exceptional divisor\end{align*} ].
Too many primes with \begin{align*}\[supersingular reduction\end{align*} ] implies \begin{align*}\[CM\end{align*} ]. Primes are supersingular about half of the time.
Open image theorems: not known for abelian varieties in general.
\begin{align*}\[Seminars and Talks/2021-04-29 The Galois Action on Symplectic K Theory\end{align*} ]
\begin{align*}\[Seminars and Talks/2021-04-29_2 Yves Andre On the canonical, fpqc and finite topologies\end{align*} ]
\begin{align*}\[Seminars and Talks/2021-04-29_3 Ribet Class groups and Galois representations\end{align*} ]
Some random notes: #todo
-Working out relative homology, an example: 
Chain of implications for module properties: 
Definitions of common matrix groups: 
Good example of exact triangles: 
Manifolds from the sheaf perspective, a reference: 
Reference: paper on “constructive” algebraic topology J. Rubio, F. Sergeraert / Bull. Sci. math. 126 (2002) 389-412 403
Many constructions in algebraic topology can be organized as solutions of fibration problems.
What are \begin{align*}\[Quillen equivalence\end{align*} ]? These need to preserve the \begin{align*}\[model structure\end{align*} ] on each side presumably.
More fundamental: how should one prove an \begin{align*}\[equivalence of categories\end{align*} ] in general?
Finding \begin{align*}\[adjunction\end{align*} ] is usually easy, because checking isomorphisms on hom sets is concrete.
If you just have a random functor, does it even have right or left adjoints in general? There must be theorems about this. See \begin{align*}\[adjoint functor theorem\end{align*} ].
What is the \begin{align*}\[Stiefel manifold\end{align*} ]?
Description of a certain wrapped \begin{align*}[Fukaya category](#fukaya-category)\end{align*} \({\mathcal{O}}\): take the objects to be (Lagrangian) embedded curves, the morphisms are the graded abelian groups \(\hom_{\mathcal{O}}\mathrel{\vcenter{:}}=\qty{\bigoplus_{L_0 \pitchfork L_1} {\mathbb{Z}}/2{\mathbb{Z}}, {{\partial}}}\) where \({{\partial}}\) is given by counting holomorphic strips, localize along small isotopies.
Add to Algebra qual review doc #todo
An ideal \({\mathfrak{p}}\) is prime iff \(JK \subset {\mathfrak{p}}\implies J \subset {\mathfrak{p}}\) or \(K\subset {\mathfrak{p}}\).
A ring is a domain iff the ideal \((0)\) is prime.
Inductively, if \({\mathfrak{p}}\) contains a product of ideals then it contains one of them.
Maximal ideals are prime, since \({\mathfrak{m}}\) maximal implies that \(R/{\mathfrak{m}}\) is a field.
A ring is local iff it has a unique maximal ideal \({\mathfrak{m}}\).
An element \(e\) is idempotent iff \(e^2 = e\).
An \(R{\hbox{-}}\)algebra \(S\) is a ring \(S\) and a homomorphism \(\alpha:R \to S\).
Every ring is a \({\mathbb{Z}}{\hbox{-}}\)algebra in a unique way.
The most interesting commutative algebras are \(S/I\) where \(S = k[x_1, \cdots, x_n]\) for \(k\) a field, \({\mathbb{Z}}\), or the \begin{align*}\[localization\end{align*} ] of a ring at a prime ideal.
See \begin{align*}\[Milnor K theory\end{align*} ]
An appearance of Milnor \(K_2\) in the wild:
How Milnor K-theory shows up in number theory: a conjecture by Tate and Birch: 
\begin{align*}\[modular form\end{align*} ] yield 2-dimensional \begin{align*}\[Galois representations\end{align*} ], and there is a classification theorem:
Deligne-Serre Theorem: 
Tags: #idle_thoughts
The \begin{align*}\[representation ring\end{align*} ] \(R(G)\): the free \({\mathbb{Z}}{\hbox{-}}\)module on isomorphism classes of irreducible \begin{align*}\[Representation theory\end{align*} ].
\begin{align*}\[equivariant cohomology\end{align*} ]? {#what-is-equivariant-cohomology}
Some uses:
Calculate number of \begin{align*}\[rational curve\end{align*} ] in a \begin{align*}\[quintic\end{align*} ] \begin{align*}\[threefold\end{align*} ] (Kontsevich 1995)
Calculate characteristic numbers of a compact \begin{align*}\[homogeneous space\end{align*} ] (Tu 2010)
Derive \begin{align*}\[Gysin formula\end{align*} ] for \begin{align*}\[examples of fiber bundles\end{align*} ] whose fibers are homogeneous spaces (Tu 2011)
Calculate integrals over manifolds as sums over fixed points: \begin{align*}\[Gauss-Bonnet\end{align*} ] and \begin{align*}\[Hopf index theorem\end{align*} ].
What is the \begin{align*}\[Homotopy quotient\end{align*} ]?
If \(G\curvearrowright M\) with \(G\) a compact connected \begin{align*}\[Lie group\end{align*} ], Cartan constructs a chain complex from \(M, {\mathfrak{g}}\).
Is this not precisely the \begin{align*}\[Borel construction\end{align*} ]?
\begin{align*}\[classifying spaces\end{align*} ] : \({\mathbf{B}}S^1 = {\mathbb{CP}}^{\infty}\)
https://www.ams.org/publications/journals/notices/201711/rnoti-p1300.pdf
Manifolds are the place to do differential calculus, \begin{align*}\[scheme\end{align*} ] are the place to do algebra by finding zeros of functions.
\begin{align*}\[Closed point\end{align*} ] : of the form \(V(S) \mathrel{\vcenter{:}}=\left\{{ q\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}q\supseteq S}\right\}\)
\begin{align*}\[homotopy colimit\end{align*} ] via Diagrams {#notes-on-homotopy-colimit-via-diagrams}
http://mathieu.anel.free.fr/mat/doc/Anel-Semiomaths-HomotopyColimit.pdf
Hocolims are \begin{align*}\[infinity groupoids\end{align*} ], equivalently \begin{align*}\[homotopy type\end{align*} ].
There is a functor \(\pi_0: { \underset{\infty}{ \mathsf{Grpd}} }\to {\mathsf{Set}}\).
\begin{align*}\[2021-04-23 Advice on research and problems\end{align*} ]
Setting goals: SMART. Doesn’t work for research though!
\begin{align*}\[attachments/image_2021-04-23-15-54-19.png\end{align*} ]
Make lists, and habitually review/revise/plan.
\begin{align*}\[attachments/image_2021-04-23-15-55-40.png\end{align*} ]
I really like the “keeping a problem list” idea.
Don’t be ashamed to ask people if they have problems you can work on.
\begin{align*}\[group cohomology\end{align*} ] in \begin{align*}\[homotopy theory\end{align*} ]? {#group-cohomology-in-homotopy-theory}
Tags: #idle_thoughts
Thinking about the link between group cohomology and homotopy theory: if I have a SES \begin{align*} 0\to A \to B \to C \to 0 ,\end{align*} should one apply a functor like \(K({-}, 1)\)? Is this actually a functor…? We definitely get spaces \(K(A, 1)\) and \(K(B, 1)\), for example, and there must be an induced map between them. Want to make precise what it means to get a SES like this: \begin{align*} 0 \to K(A, 1) \to K(B, 1) \to K(C, 1) \to 0 .\end{align*} One would kind of want this to be part of a \begin{align*}\[fiber sequence\end{align*} ] I guess. But we’re in \({\mathsf{Top}}\) anyway, so there’s no real issue with just doing \begin{align*}\[fibrant and cofibrant objects\end{align*} ],.
Maybe the “right” think to do here is to actually take a classifying \begin{align*}\[groupoid\end{align*} ] (?), which must be some functor like \({\mathbf{B}}: {\mathsf{Grp}}\to {\mathsf{Grpd}}\). Surely this is some known thing. But then what is an “exact sequence of groupoids”…? \begin{align*} 0 \to {\mathbf{B}}A \to {\mathbf{B}}B \to {\mathbf{B}}C \to 0 .\end{align*}
Also, why should such a functor be an exact? It’d kind of be more interesting if it weren’t. Say it’s right-exact, then how might you make sense of \(\mathop{\mathrm{{\mathbb{L} }}}{\mathbf{B}}({-})\)? I think this just needs a model category structure on the source, although it seems reasonable to expect that \({\mathsf{Grpd}}\) would have some simple model structure.
\begin{align*}[l-adic representations](#l-adic-representations)\end{align*}
Try computing things like \({ \mathsf{Gal}} ({\mathbb{Q}}( \zeta_3, \sqrt{3})\).
There’s some way to check orders of Galois groups using \begin{align*}\[valuation\end{align*} ]..?
See Néron-Ogg-Shafarevich criterion: \begin{align*}\[good reduction\end{align*} ] iff \begin{align*}\[Inertia\end{align*} ] acts trivially, or \begin{align*}\[semistable reduction\end{align*} ] iff inertia acts \begin{align*}\[unipotently\end{align*} ].
Always have quasi-unipotently, so eigenvalues roots of unity.
\begin{align*}\[Galois representations\end{align*} ] at different primes are related, using local info at a few primes to get global info at all primes.
My main question: does introducing derived stacks somehow make some computation easier?
Integrating over a \begin{align*}\[fundamental class\end{align*} ] :
Appearance of \begin{align*}\[Calabi-Yau\end{align*} ] in \begin{align*}\[Physics\end{align*} ]
Mirror symmetry of CYs:
The major types of “moduli” style invariants
Why care about \begin{align*}\[coherent sheaves\end{align*} ]?
\begin{align*}\[Donaldson-Thomas invariants\end{align*} ] are supposed to relate to \begin{align*}\[Gromov-Witten invariants\end{align*} ] :
Niceness of spaces:
We can’t prove the \begin{align*}\[Tate conjecture\end{align*} ]? I guess this is an arithmetic analog of the \begin{align*}\[Hodge conjecture\end{align*} ]. Serre’s book calls some isomorphism the Tate conjecture and says it’s proved though.
Pithy explanation of a \begin{align*}\[derived scheme\end{align*} ] : a space which can be covered by Zariski opens \(Y\cong \operatorname{Spec}A^*\) where \(A\in {\mathsf{cdga} }_{k}\).
\begin{align*}\[scheme\|schemes\end{align*} ] and \begin{align*}\[stack\|stacks\end{align*} ] can be very singular.
\begin{align*}\[Derived schemes\end{align*} ] and \begin{align*}\[derived stacks\|derived stacks\end{align*} ] act a bit like smooth, nonsingular objects.
Derived modular stacks of \begin{align*}\[quasicoherent sheaf\|quasicoherent sheaves\end{align*} ] over \(X\) remember the entire \begin{align*}\[deformation theory\end{align*} ] of sheaves on \(X\).
#seminar_notes #blog #prisms
One can take \begin{align*}\[etale cohomology\end{align*} ] of varieties, and later refine to schemes, and thus take it for the base field even when it’s not algebraically closed and extract arithmetically interesting information.
\begin{align*}\[prismatic cohomology\end{align*} ], meant to relate a number of other cohomology theories
\begin{align*}\[Prism\end{align*} ] : a pair \((A, I)\) where \(A\) is a commutative ring with a derived Frobenius lift \(\phi:A\to A\), i.e. a \(\delta{\hbox{-}}\)structure.
Fix a scheme and study prisms over it. Need these definitions to have stability under base-change.
Examples:
Prismatic \begin{align*}\[site\end{align*} ] : fix a base prism \((A, I)\) for \(X\) a \(p{\hbox{-}}\)adic \begin{align*}\[formal scheme\end{align*} ] over \(A/I\). Define \begin{align*} (X/A)_\prism = \left\{{ (A, I) \to (B, J) \in \mathop{\mathrm{Mor}}(\mathsf{Prism}), \operatorname{Spf}(B/J) \to X \text{ over } A/I }\right\} ,\end{align*} topologized via the \begin{align*}\[flat topology\end{align*} ] on \(B/J\).
There is a \begin{align*}\[structure sheaf\end{align*} ] \({\mathcal{O}}_\prism\) where \((B, J) \to B\). Take \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma\), which receives a Frobenius action, to define a cohomology theory. Why is this a good idea?
Absolute prismatic sites: for \(X\in {\mathsf{Sch}}(p{\hbox{-}}\text{adic})\), define \begin{align*} X_\prism \mathrel{\vcenter{:}}=\left\{{ (B, J) \in \mathsf{Prism},\, \operatorname{Spf}(B/J) \to X }\right\} .\end{align*} Take \begin{align*}\[sheaf cohomology\end{align*} ] to obtain \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism(X) \mathrel{\vcenter{:}}=\mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_\prism, {\mathcal{O}}_\prism) {\circlearrowleft}_\phi\).
The category \(\mathsf{Prism}\) doesn’t have a \begin{align*}\[final object\end{align*} ], so has interesting cohomology. Relates to \begin{align*}\[Algebraic K theory\end{align*} ] of \({\mathbb{Z}}_p\)?
Questions: let \(X_{/{\mathbb{Z}}_p}\) be a smooth formal scheme.
Bhatt and Lurie: found a stacky way to understand the absolute prismatic site of \({\mathbb{Z}}_p\). Drinfeld found independently.
Construction due to Simpson: take \(X\in {\mathsf{Var}}({\mathsf{Alg}})\), define a de Rham presheaf \begin{align*} X_{\mathrm{dR}}: {\mathsf{Alg}_{\mathbb{C}} }^{\operatorname{fp}} &\to {\mathsf{Set}}\\ R &\mapsto X(R_{ \text{red} }) .\end{align*}
Def: Cartier-Witt stack, a.k.a. the prismatization of \({\mathbb{Z}}_p\)
Plug in a \(p{\hbox{-}}\)nilpotent ring \(R\) to extract all (derived) prism structure on \(W(R)\).
Prisms aren’t base-change compatible without the derived part.
This is a \begin{align*}\[groupoid\end{align*} ].
An explicit presentation: \(\mathsf{WCart}_0(R)\) are distinguished \begin{align*}\[Witt Vectors\end{align*} ] in \(W(R)\). Given by \([a_0, a_1, \cdots ]\) where \(a_0\) is nilpotent and \(a_1\) is a unit. This is a formal affine scheme. \(\mathsf{WCart}= \mathsf{WCart}_0 / W^*\) is a presentation as a \begin{align*}\[stack quotient\end{align*} ].
Start by understanding its points, suffices to evaluate on fields of characteristic \(p\).
If \(k\in \mathsf{Field}(\mathsf{Perf})_{\operatorname{ch. p}}\), \(\mathsf{WCart}(k) = \left\{{ {\operatorname{pt}}}\right\}\), with the point represented by \((W(k), ?)\).
Analogy to understanding \begin{align*}\[Hodge-Tate cohomology\end{align*} ]. Similar easy locus in this stack.
Take 0th component of distinguished \begin{align*}\[Witt Vectors\end{align*} ] to get a diagram
The bottom-left is this \begin{align*}\[Hodge-Tate stack\end{align*} ]
Now has a better chance of being an \begin{align*}\[algebraic stack\end{align*} ] instead of a \begin{align*}\[formal stack\end{align*} ]. Bottom arrow kills the formal direction.
Will be \begin{align*}\[classifying stack\end{align*} ] of a \begin{align*}\[group scheme\end{align*} ] : need to produce a point and take automorphisms.
Take the distinguished element \(V(?) \in W({\mathbb{Z}}_p)\). Produces a map \begin{align*} \operatorname{Spf}({\mathbb{Z}}_p) \xrightarrow{\pi_{\operatorname{HT}}} \mathsf{WCart}^{\operatorname{HT}} .\end{align*}
Upshot: \(\mathsf{WCart}^{\operatorname{HT}}= {\mathbf{B}}W^* [F]\) is a classifying stack. \begin{align*}\[quasicoherent sheaves\end{align*} ] on the left and representations of the (classifying stack of the) group scheme on the right. I.e. \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart}^{\operatorname{HT}}) = \mathop{\mathrm{{\mathbb{R} }}}(W^*[F])\).
Teichmüller lift yields a \({\mathbb{Z}}/p\) grading on the LHS.
Something about Deligne-Illusie? \begin{align*}\[Hodge-to-deRham degeneration\end{align*} ]
Upshot: a \begin{align*}\[divisor\end{align*} ] inside is easy to understand.
Fact: \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart})\) are equivalent to \begin{align*} \varprojlim_{(A, I)\in \mathsf{Prism}} { \mathsf{D} }_{(P, I)-?}(A) .\end{align*}
Diffracted Hodge cohomology: let \(X\in {\mathsf{Schf}}_{{\mathbb{Z}}_p}\). Get a prismatic structure sheaf using the assignment \((A, I) \to \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism \qty{ (X\otimes A/I) / A}\).
Heuristic: \(\operatorname{Spec}{\mathbb{Z}}_p\) should be 1-dimensional over something.
Get an absolute comparison: \(\operatorname{cohdim}\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism (X) \leq d+1\) where \(d = \operatorname{reldim}X_{/{\mathbb{Z}}_p}\).
There is a deRham comparison: \begin{align*} X_{{\mathbb{F}_p}}^* H_\prism(X) \cong \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) .\end{align*}
There is a Hodge-Tate comparison: the object \(H_\prism(X)\) restricted to \(\mathsf{WCart}^{\operatorname{HT}}\) has an increasing filtration with \({\mathsf{gr}\,}_i = \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X, \Omega^i_X)[-i]\).
Combine these comparisons to get \begin{align*}\[Deligne-Illusie\end{align*} ] : if \(\operatorname{reldim}X < p\), then \begin{align*} \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) \cong \bigoplus_{i} \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_{{\mathbb{F}_p}}, \Omega^i[-i]) .\end{align*} Get a lift to characteristic zero, yields Hodge-to-deRham degeneration there.
An \(F{\hbox{-}}\)crystal on \(X_\prism\) is a vector bundle \(\mathcal{E} \in { \mathsf{Vect} }(X_\prism, {\mathcal{O}}_\prism)\)? Plus some extra data.
Infinite tensor product: \begin{align*} I_\prism \otimes F^* I_\prism \otimes(F^2)^* I_\prism \otimes\cdots .\end{align*} Converges to some object \({\mathcal{O}}_\prism \left\{{ 1 }\right\} \in {\operatorname{Pic}}(X_\prism, {\mathcal{O}}_\prism )\), twisted? Yields isomorphism of sheaves after inverting \(I_\prism\), \begin{align*} F^* {\mathcal{O}}_\prism \left\{{ 1 }\right\} \cong I_\prism^{-1}\otimes{\mathcal{O}}_\prism \left\{{ 1 }\right\} .\end{align*}
Some analog of \begin{align*}\[Artin-Schreier\end{align*} ] here, taking fixed points?
There is a natural functor from \(F{\hbox{-}}\)crystals on \(X\) to local \({\mathbb{Z}}_p\) systems on a geometric fiber \(X_?\)?
Main theorem: produces \begin{align*}\[descent\end{align*} ] data, uses work on Beilinson fiber sequence (Benjamin Antieau, Morrow, others?)
Can say \begin{align*} H^i(\Delta_{{\mathbb{Z}}_p}) = \begin{cases} {\mathbb{Z}}_p & i=0 \\ \prod_{{\mathbb{N}}} {\mathbb{Z}}_p & i=1. \end{cases} \end{align*} Can compute using \begin{align*}\[topological Hochschild homology\end{align*} ]? \(\pi_{-1}( {\operatorname{TP}}({\mathbb{Z}}_p) )\) is where the \(i=1\) part comes from.
Prismatic is filtered by things that look like Hodge-Tate
Absolute = arithmetic (take eigenspaces, related to \begin{align*}\[motivic cohomology\end{align*} ], relative = geometric?
Similar to situation in \begin{align*}\[Étale cohomology\end{align*} ] : need absolute and relative to compute either.
\begin{align*}\[triangulated categories\end{align*} ] and \begin{align*}\[derived category\end{align*} ] important? {#why-are-triangulated-categories-and-derived-category-important}
\begin{align*}\[Hill-Hopkins-Ravenel\end{align*} ] {#a-roadmap-to-hill-hopkins-ravenel}
\begin{align*}\[Lurie stuff\end{align*} ] Content {#some-lurie-stuff-content}
Lurie’s Seminar on Algebraic Topology
Lurie’s Topics in Geometric Topology
\begin{align*}\[topological Hochschild homology\|THH\end{align*} ] and \begin{align*}\[K-Theory\end{align*} ] {#the-relationship-between-topological-hochschild-homologythh-and-k-theory}
Some remarks on \({\operatorname{THH}}\) and \(K{\hbox{-}}\)Theory, no clue what the original source was:
\begin{align*}\[Algebraic K theory\end{align*} ] is hard, using \begin{align*}\[topological Hochschild homology\end{align*} ] somehow makes computations easier.
\(K{\hbox{-}}\)theory says something about \begin{align*}\[vector bundles\end{align*} ], \begin{align*}\[topological Hochschild homology\end{align*} ] describes \begin{align*}\[monodromy\end{align*} ] of vector bundles around infinitesimal loops
For \(X\) a nice \begin{align*}\[scheme\end{align*} ], take \(LX\) the derived \begin{align*}\[free loop space\end{align*} ] : the \begin{align*}[derived stacks](#derived-stacks)\end{align*} \(\mathop{\mathrm{Maps}}_{ \operatorname{DSt}}(S^1, X)\).
Identify \({\operatorname{THH}}(X) \xrightarrow{\sim} {\mathcal{O}}(LX)\) (global functions)
\begin{align*}\[Dennis trace\end{align*} ] : a comparison \(K(X) \to {\operatorname{THH}}(X)\), takes \(E\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }\) to the canonical monodromy automorphism of the pullback of \(E\) to \(LX\)
Traces are \(S^1{\hbox{-}}\) \begin{align*}\[equivariant\end{align*} ] because loops! Just equip \(K(X)\) with the trivial \(S^1\) action.
Take \begin{align*}\[homotopy fixed points\end{align*} ] to get something smaller than \({\operatorname{THH}}\): \({\operatorname{THC}}^-\), topological negative cyclic homology
Dennis trace is invariant under all covering maps of circles, even multisheeted
Theorem of Dundas-Goodwillie-McCarthy: whenever \(A\to A'\) is a nilpotent extension of connective \begin{align*}\[ring spectra\end{align*} ], \(K(A') \xrightarrow{\sim} K(A) { \underset{\scriptscriptstyle {{\operatorname{TC}}(A)} }{\times} } {\operatorname{TC}}(A')\)
\begin{align*}\[Eilenberg-MacLane spaces\end{align*} ] {#eilenberg-maclane-spaces}
\begin{align*}\[stacks\end{align*} ]? {#why-care-about-stacks}
Why should I care about \begin{align*}[derived stacks](#derived-stacks)\end{align*} ?
Note from Arun: one can get \begin{align*}\[Topological modular forms\|TMF\end{align*} ] and \begin{align*}\[Topological modular forms\|tmf\end{align*} ] along with their ring structures without doing \begin{align*}\[Obstruction Theory\end{align*} ]
\begin{align*}\[Lie algebra cohomology\end{align*} ] {#homotopy-theory-is-connected-to-lie-algebra-cohomology}
\begin{align*}\[schemes\end{align*} ] and \begin{align*}\[class field theory\end{align*} ] {#schemes-and-class-field-theory}
See \begin{align*}\[Arithmetic schemes\end{align*} ]
Idele group for arithmetic schemes
-Actual class group for schemes
Wiesend’s finiteness theorem is one of the strongest and most beautiful results in higher \begin{align*}\[Global class field theory\end{align*} ]?
The main aim of higher \begin{align*}\[Global class field theory\end{align*} ] is to determine the abelian fundamental group \(\pi_1^{{\operatorname{ab}}}(X)\) of a regular arithmetic scheme \(X\), i.e. of a connected \begin{align*}\[regular scheme\end{align*} ] \begin{align*}\[separated scheme\end{align*} ] scheme \begin{align*}\[flat morphism\end{align*} ] and of \begin{align*}\[finite type\end{align*} ] over \({\mathbb{Z}}\), in terms of an arithmetically defined \begin{align*}\[class groups\end{align*} ] \(C(X)\).
Fundamental theorem of class field theory?
#modular_forms
Modular forms can be defined as functions \({\mathbb{H}}\to {\mathbb{C}}\) satisfying weak \(\Gamma{\hbox{-}}\)invariance.
Also sections of a bundle: the \begin{align*}\[modular curve\end{align*} ].
\begin{align*}\[weight of a modular form\end{align*} ] : refers to growth rates of these functions.
Weird fact: \(M_1\) is one-dimensional, but for \(g\geq 2\) we have \(\dim M_g = 3g-3\)
Higher genus generalizations come not from a Teichmüller cover \(T_g \twoheadrightarrow M_g\) or \(M_g\), no one seems to care about those though.
\({\mathbb{H}}_S^g/{\mathsf{Sp}}(2g; {\mathbb{Z}})\) is somehow a model for \begin{align*}\[moduli stack of abelian varieties\end{align*} ], \(M_g\) embeds as a variety since we have the \begin{align*}\[Jacobian\end{align*} ]
\begin{align*}\[Hodge bundle\end{align*} ] : rank \(g\) over \(M_g\), fibers over isomorphism classes are \(H^0(X, K_X)\) where \(K\) is the \begin{align*}\[canonical bundle\end{align*} ], then take the determinant bundle.
#advice
Reminding myself of Ravi’s “Three Things” exercise.
\begin{align*}\[Three Things Exercise\#Process for talks\end{align*} ]
#seminar_notes
What is the \begin{align*}\[universal curve\end{align*} ] over \(M_g\)?
What is the \begin{align*}\[tautological ring\end{align*} ], and how does it relate to the \begin{align*}\[Chow ring\end{align*} ]?
What are the \begin{align*}\[kappa classes\end{align*} ]?
Question: when does \(A^*(M_g) = R^*(M_g)\)?
The integral Chow ring is very unknown. We know it only very recently for \(M_2, \mkern 1.5mu\overline{\mkern-1.5muM_2\mkern-1.5mu}\mkern 1.5mu\). Compare to rational Chow rings: we know them as quotients of polynomial rings for \(M_g, g\leq 23\).
Can stratify \(M_g\) by \begin{align*}\[gonality\end{align*} ].
What are \begin{align*}\[hyperelliptic\end{align*} ] curves?
I definitely have known this at several points in time, yeesh.
Look at \begin{align*}\[Hurwitz space\end{align*} ] : \begin{align*}\[moduli space\end{align*} ] of degree \(n\) covers of \({\mathbb{P}}^1\) by smooth genus \(g\) \begin{align*}\[curves\end{align*} ]? At least this specific one is.
Reference: Chao Li, “Beilinson-Bloch conjecture for unitary Shimura varieties.” Priinceton/IAS NT Seminar
What is the \begin{align*}\[Beilinson Bloch conjecture\end{align*} ]?
Beilinson-Bloch conjecture: generalizes the \begin{align*}\[Birch and Swinnerton-Dyer conjecture\end{align*} ].
What are higher \begin{align*}\[Chow ring\end{align*} ] ? What do they generalize?
What is an \begin{align*}\[adele\end{align*} ]? What is an \begin{align*}\[adele\end{align*} ] point?
I should also review what a \begin{align*}\[place\end{align*} ]really is. Definitely what it means to be an \begin{align*}\[Archimedean place\end{align*} ]. Also double-check the \(v\divides \infty\) notation.
What is an \begin{align*}\[automorphic representation\end{align*} ]?
See \begin{align*}\[Gross-Zagier\end{align*} ] formula.
What is a \begin{align*}\[modular curve\end{align*} ]?
What is a \begin{align*}\[Heegner divisor\end{align*} ] for some \begin{align*}\[imaginary quadratic field\end{align*} ] over \({\mathbb{Q}}\) and why can one use the theory of \begin{align*}\[complex multiplication\end{align*} ] to get it defined over other fields?
Gotta learn \begin{align*}\[modular form\end{align*} ]. They can take values in the complexification of a \begin{align*}\[Mordell-Weil group\end{align*} ]? Also need to know something about \begin{align*}\[Hecke operator\end{align*} ].
What is a \begin{align*}\[Shimura variety\end{align*} ]?
What is a theta series? Something here called an arithmetic theta lift, where some pairing form generalizes Gross-Zagier (?). See Beilinson-Bloch height maybe?
I should read a lot more about \begin{align*}\[Chow ring\|Chow groups\end{align*} ].
What is \begin{align*}\[Betti cohomology\end{align*} ]?
Why is proving that something is \begin{align*}\[modular form\end{align*} ] a big deal?
Look for the Kudla Program in arithmetic geometry, and \begin{align*}\[Kudla-Rapoport conjecture\end{align*} ].
Comment by Peter Sarnak: BSD was first checked numerically for CM elliptic curves!
What is a Siegel \begin{align*}\[Eisenstein series\end{align*} ]? Or even just an Eisenstein series.
See \begin{align*}\[Néron-Tate height\end{align*} ] pairing? Seems like these BB heights can only really be computed locally, then you have to sum over places.
What are the \begin{align*}\[Standard conjectures\end{align*} ]?
Main formula and big theorem:
Seems that we know a lot about the LHS, the right-hand side is new. We don’t know nondegeneracy of the RHS, for example, e.g. the pairing vanishing implying the cycle is zero.
See \begin{align*}\[Tate conjecture\end{align*} ].
Comment from Peter Sarnak: we know very little about where \(L\) functions vanish, except for \(1/2\).
Need to do \begin{align*}\[Resolution of singularities\end{align*} ] when you don’t have a “regular” (integral?) model.
Paper recommended by Juliette Bruce: https://arxiv.org/pdf/2003.02494.pdf
Jonathan Love! Shows some cool consequences of the \begin{align*}\[Beilinson Bloch conjecture\end{align*} ], primarily a 2-parameter family of \begin{align*}\[elliptic curve\end{align*} ] where the image \({\operatorname{CH}}^1(E_1)_0 \otimes{\operatorname{CH}}^1(E_2)_0 \to {\operatorname{CH}}^2(E_1 \times E_2)\) is finite. BB predicts this is always finite when defined over \(k\) a \begin{align*}\[number field\end{align*} ].
I should remind myself what \begin{align*}\[local fields\end{align*} ] and \begin{align*}\[global field\|global fields\end{align*} ] are.
What is the \begin{align*}\[Bloch-Kato\end{align*} ] conjecture? What does it predict for \(L{\hbox{-}}\)functions?
What is the \begin{align*}\[Birch and Swinnerton-Dyer conjecture\end{align*} ]? What does this predict for \begin{align*}\[L function\|L functions\end{align*} ]?
What is a \begin{align*}\[regulator\end{align*} ]?
What is the \begin{align*}\[height pairing\end{align*} ]?
#research_projects
Some motivation for
\begin{align*}\[K3 Surfaces\end{align*}
] : Fermat hypersurfaces \(\sum x_i ^k\) for some fixed \(k\).
Look for \({\mathbb{Q}}{\hbox{-}}\)points, since by homogeneity the denominators can be scaled out to get \({\mathbb{Z}}{\hbox{-}}\)points
\begin{align*}\[Falting's theorem\end{align*} ] : for a \begin{align*}\[curves\end{align*} ] \(C\) with \(g(C) \geq 2\), the number of \begin{align*}\[rational points\end{align*} ] is finite, i.e. \({\sharp}C({\mathbb{Q}}) < \infty\).
Diagonal hypersurfaces \(x_0^k + \cdots + x_n^k = 0\).
\begin{align*}\[Calabi-Yau\end{align*}
] when \(k=n+1\) (maybe a bound instead..?), sharp change in behavior of finiteness of rational points at this threshold.
#seminar_notes
\begin{align*}\[Dehn surgery\end{align*} ] : remove a tubular neighborhood of a knot, i.e. a solid torus, glue back in by some diffeomorphism of the boundary.
\begin{align*}\[L Space conjecture\end{align*} ] simplest \begin{align*}\[Heegard-Floer homology\end{align*} ], rank of \(\operatorname{HF}\) equals cardinality of \(H_{\mathrm{sing}}\).
Left-orderability on groups: a total order compatible with the group operation. Torsion groups can’t be LO: \(x>1 \implies 1 = x^n > \cdots > x > 1\).
\begin{align*}\[taut foliation\end{align*} ] : a geometric condition. Admits a decomposition into leaves where a simple closed curve intersects each transversally?
\begin{align*}\[fibred\end{align*} ] 3-manifolds: take \(\Sigma \times I\) for \(\Sigma\) a surface, glue the top and bottom by some diffeomorphism \(\phi: \Sigma: {\circlearrowleft}\).
Osvath-Szabo: admitting a taut foliation implies being a non-\(L{\hbox{-}}\)space. Is the converse true?
Interesting \begin{align*}\[knot invariants\end{align*} ] : \(\tau, s, g_4(K), \sigma\). Also the Jones, Conway, Alexander polynomials, or even just a coefficient. Note that some of these polynomials can not admit cabling formulas.
Tags: #qual_analysis
\begin{align*}\[quasiisomorphism\end{align*} ] :
See \begin{align*}\[qual review\end{align*} ].
Note from Pete: a common technique on qual Algebra problems is extracting an index 2 subgroup using the Cayley action of \(G\) on itself. These are always normal, so if this exists, \(G\) can’t be simple unless it’s order 2.
The connections between \begin{align*}\[Formal group\end{align*} ] and \begin{align*}\[chromatic homotopy theory\end{align*} ] are key. Read Quillen’s paper, J.F. Adams’ blue book, Ravenel, etc. Ravenel has some slides on Quillen’s work (good entry pt). If you really want to go in depth, I enjoyed learning from Hazewinkel’s book “Formal Groups and Applications.”
#advice #goals
Don’t set goals to be outcome-dependent. Set input goals! E.g. “Write a book that I’m proud of and I like.”
Don’t set goals that depend on factors outside of your control – hitting some metric, getting a specific award, etc.
Check in with your emotional state. It’s not always healthy to push through not feeling like doing something. This can be useful if it’s a matter of discipline, i.e. if pushing through is actually serving you well. But it can also serve you poorly in the long run by exacerbating poor mental state or leading to burnout.
It’s okay to take a break. Check in to ask yourself if it’s coming from a place of self-care or instead procrastination.
Choose to be satisfied! The story you tell yourself after the fact does not change the reality of the past. Does it serve you well to say “I didn’t do enough?” It’s okay to choose to be satisfied.
Have a mental model is the key priority, then go to rote memorization only when absolutely necessary. Remember because you understand the subject!
Use active recall for the learning process – actively test yourself as you’re reading.
See book: “Make It Stick”
The evidence is that popular techniques have low utility: rereading, highlighting, summarizing, taking notes.
Summarizing and taking notes: low utility in general. This has some marginal utility if you’re particularly skilled or trained in summarizing effectively. So not very effective, but do it if it brings you joy!
Active recall and practice testing: high utility! Not very time-intensive, doesn’t require special skills/training.
Do practice testing during each study session. Studies show 10-15% improvement. Try to ask yourself inference questions.
Studies show that those doing retrieval practice actually predict the smallest improvements and achieve the highest improvements. Very counterintuitive, trust the process!
Reading once and practice testing can be more effective than rereading 4 times, and takes less time.
Make first draft of notes with the book closed, no references.
Spaced repetition: also works on a micro scale! E.g. within a single study session.
Practice a little bit each day over a long period of time.
Extremely important: scope your subject. Know the broad outline of the course inside and out. Can scope based on actual exams.
Start from topics you don’t know as well. E.g. work backwards though lecture notes.
Try interleaved practice: don’t necessarily need to master a topic before moving on, realizing that you’ll be reviewing it several times again before it’s needed.
Idea for tracking system: put course outline by topics into a spreadsheet, then record dates of review next to each topic. Color code cells based on quality of recall.
Revision needs to be a fluid process, not worth timetabling down to each section due to different time requirements for each. Pick what makes your brain work the hardest.
Reference: Link to PDF
See \begin{align*}\[Spectral sequence\end{align*} ].
What is a \begin{align*}\[Dirichlet character\end{align*} ]?
What is a \begin{align*}\[Dirichlet L function\end{align*} ]?
\begin{align*} L(s ; \chi):=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}} =\prod_{p} \qty{ 1-\chi(p) p^{-s} }^{-1} .\end{align*}
How is a \begin{align*}\[Bernoulii number\end{align*} ] defined? Generalized Bernoulli numbers:
What is the \begin{align*}\[conductor\end{align*} ] of a \begin{align*}\[Dirichlet character\end{align*} ]?
What is the \begin{align*}\[J-homomorphism\end{align*} ]?
How is it defined in terms of \begin{align*}\[loop space\end{align*} ]?
How is it defined in terms of \begin{align*}\[framed cobordism\end{align*} ]? What is a \begin{align*}\[framed\|framing\end{align*} ]?
How is it defined in terms of \begin{align*}\[Thom space\end{align*} ]? What is a Thom space?
What is a \begin{align*}\[complex oriented cohomology theory\end{align*} ]?
What is a \begin{align*}\[uniformizer\end{align*} ]?
What is a \begin{align*}\[Kan extension\end{align*} ]? #todo
Kan extensions:
What are \begin{align*}\[derived functors\end{align*} ] in a \begin{align*}\[Model category theory\|model category\end{align*} ]?
What is a \begin{align*}\[Quillen adjunction\end{align*} ]?
What is a \begin{align*}\[Quillen equivalence\end{align*} ]?
What is an \begin{align*}\[infinity categories\|infinity category\end{align*} ]?
What is a \begin{align*}\[topological category\end{align*} ]?
What does it mean for a category to be \begin{align*}\[Enriched category\|enriched\end{align*} ]?
What is the \begin{align*}\[homotopy category\end{align*} ] of a \begin{align*}\[Model category theory\|model category\end{align*} ]?
What is the model structure on (differential graded) vector spaces? What are the fibrations, cofibrations, and weak equivalences?
What does an augmented algebra correspond to a pointed affine \begin{align*}\[scheme\end{align*} ]?
How are initial and final objects defined in \begin{align*}\[infinity categories\|infinity category\end{align*} ]?
#rational_points
\begin{align*}\[Vojta's conjecture\end{align*} ] : Gives good \begin{align*}\[height\end{align*} ] estimates, can be used to prove something is finite.
\begin{align*}\[Northcott's theorem\end{align*} ] Used in arithmetic dynamics since bounded height and bounded degree implies finite.
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The first case is a strong inversion, the second is a 2-periodic involution (given by twisting about a core torus).
One can assume that \(\tau\) is rotation about some axis.
There is an extension of \(\tau\) to \({\mathbb{B}}^4\), so define an equivariant slice surface \(\Sigma\) if \(\tau \Sigma = \Sigma\), and define an equivariant (slice?) genus as the minimal genus among such surfaces \(\tilde g_4(K)\)
Study \(\tilde g_4(K) - g_4(K)\). Boyle-Issan show this difference is unbounded for a family of periodic knots.
Prove a similar theorem: given \((K, \tau)\), define a set of numerical invariants using Floer homology which are
Produced a family of strongly invertible slice knots where \(\tilde g_4\) is unbounded.
Most (small crossing) knots admit a strong inversion.
Next: how to apply this machinery to seemingly non-equivariant things.
A slice surface \(\Sigma\) is isotopy equivariant iff \(\tau_{{\mathbb{B}}^4} \isotopic \Sigma\) rel boundary. Define isotopy equivariant genus \(\tilde{ig}_4(K)\) as the minimal genus of such \(\Sigma\).
Calculating this invariants gives a way of finding non-isotopic surfaces for \(K\).
Recent work: topologically isotopic but not smoothly isotopic surfaces.
Proving topologically isotopic: a known theorem involving equivalence of \(\pi_1\).
Theorem: produced a knot where \(\tilde{ig}_4(K) > 0\).
Does \({\mathbb{B}}^4\) actually matter here? The answer is no, can take \(\operatorname{ZHB}^4\).
A generalized isotopy equivariant surface is a triple \((W, \tau_W, \Sigma)\) where
Another application: let \(\Sigma, \Sigma'\) be two slices surfaces in \({\mathbb{B}}^4\) for \(K\). Interpolate: take \(\Sigma = \Sigma_0 \to \Sigma_1 \to \cdots \to \Sigma_n = \Sigma'\) where each arrow is a stabilization or destabilization or isotopy rel \(K\). How many arrows are needed? Define this as \(M_{{~\mathrel{\Big\vert}~}}(\Sigma, \Sigma')\), the stabilization number.
Theorem: for any \(m\), produce a knot \(J_m\) with two slice disks with stabilization distance exactly \(m\).
Theorem: if \((K, \tau)\) is strongly invertible slice and \(\Sigma\) is any slice disk for \(K\), then \(M_{{~\mathrel{\Big\vert}~}}(\Sigma, \tau_{{\mathbb{B}}^4} \Sigma) \geq \cdots\), some function of the numerical invariants.
These all induce maps on \(\CFK(K)\), where the \(\tau\) action induces a \(\tau\) action on \(\CFK(K)\). Isotopy equivariant knot cobordisms \(K_1\to K_2\) induce \(\tau{\hbox{-}}\)equivariant maps \(\CFK(K_1) \to \CFK(K_2)\) in the sense that this commutes with the two different \(\tau\) actions on either side.
Can use this to find knots that are concordant but not equivariantly concordant by using algebraic restrictions on bigraded \(\CFK(K)\)
Doing this with higher order diffeomorphisms: the roadblock is defining \(\operatorname{HF}\) mod \(p\)!
A nice modern intro to homotopy theory: https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/intro-homoto.pdf
Quotients are colimits:
Geometric realization as a \begin{align*}\[coend\end{align*} ]
Homotopy fibers:
Homotopy cofiber:
Spectra as a presentable \begin{align*}\[infty-category\end{align*} ]
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Refs: ?
Hector Pasten, UGA NT seminar.
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Torelli: the map sending a curve to its Jacobian is an injection on points.
Intermediate Jacobian: introduce to prove irrationality of cubic threefolds. An abelian variety the parameterizes degree zero cycles in dimension 1, up to rational equivalence.
Relationship between complex projective and geometry and symplectic topology: Kähler manifolds.
Abouzaid: interesting results about symplectic topology of Hamiltonian fibrations over the 2-sphere, and their consequences for smooth projective maps over the projective line.
The Grothendieck group of mixed Hodge modules, which enhances the Grothendieck group of \(G{\hbox{-}}\)modules.
A motivic semiorthogonal decomposition is the decomposition of the derived category of a quotient stack \begin{align*}X/G\end{align*} into components related to the “fixed-point data.” They represent a categorical analog of the Atiyah-Bott localization formula in equivariant cohomology, and their existence is conjectured for finite G
Can define curvature and 2nd fundamental form for algebraic varieties?
Invariants like HOMFLY: invariants of quantum matrices
consider the stack of representations, its inertia stack and the nilpotent version of the inertia stack.
Hurwitz spaces H_{k,g}, parametrizing degree k, genus g covers of P^1
Kobayashi–Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite–Einstein metric if and only if the bundle is slope polystable
predicted that given two vector bundles V_1, V_2 whose first Chern classes both vanish and whose second Chern classes agree, the resulting line bundles Thom(V_1) and Thom(V_2) should agree in Pic(Ell_G(X)).
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Refs: ?
Hannah Turner, GT: Branched Cyclic Covers and L-Spaces
Two main constructions for 3-manifolds: Dehn surgery and branched cyclic covers
Idea: \(C_n\curvearrowright M\), take quotient to get an \(n{\hbox{-}}\)fold covering map away from a branch locus (usually a knot or link).
Given a knot \(K\hookrightarrow S^3\), can produce a canonical cyclic branched cover for any \(n\), \(\Sigma_n(K)\).
Dehn surgeries: classified by \(p/q \in {\mathbb{Q}}\).
Fact: \(\dim_{{\mathbb{F}}_2} \widehat{\operatorname{HF}}(M) \geq \# H_1(M; {\mathbb{Z}})\) unless it’s infinite, in which case we set the RHS to zero. We say \(M\) is an \(L{\hbox{-}}\)space if this is an equality.
Conjecture: non \(L{\hbox{-}}\)space if and only if admits a co-oriented taut foliation (decomposition into surfaces) iff \(\pi_1\) is left orderable.
Q: push through local system correspondence, what does this say about reps \(\pi_1\to G\)..? Or local systems..?
We know foliation \(\implies\) non \(L{\hbox{-}}\)space, the other directions are all wide open.
Diagrams for knots: boxes with numbers are half-twists, sign prescribes directions.
Which branched covers of knots are \(L{\hbox{-}}\)spaces?
Nice trick: quotient by a \(C_2\) action to make it a double branched cover \(X\to X/C_2\), and find an \(n{\hbox{-}}\)fold branched cover \(\tilde X\to X\). Then take an \(n{\hbox{-}}\)fold branched cover \(\tilde{X/C_2} \to X/C_2\) and then its 2-fold branched cover will be \(\tilde X \to \tilde{X/C_2}\).
Weakly quasi-alternating \(K\): \(\Sigma_2(K)\) is an \(L{\hbox{-}}\)space.
There are tools for showing the \(\pi_1\) you get here are not left-orderable. Showing left-orderability: fewer tools, need representation theory.
Generalized \(L{\hbox{-}}\)space: \(L{\hbox{-}}\)spaces for \(S^1\times S^2\).
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Differential forms on (derived) \begin{align*}\[stacks\end{align*} ]:
What is non-commutative geometry?
Category of singularities:
Bloch’s conductor conjecture:
On harmonic bundles:
Existence of spin and string structures: kind of like applying a functor to the Whitehead tower and asking for sections of the image tower:
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UGA AG Seminar: Eloise Hamilton?
GIT: \(G\) a reductive group (trivial unipotent radical), \(G\curvearrowright X\) a projective variety, a lift of the action to an ample line bundle \({\mathcal{L}}\to X\) so that \(G\) “acts on functions on \(X\).”
Making GIT work more generally in non-reductive settings: adding a \({\mathbb{G}}_m\) grading seems to fix most issues!
Definition: \(H = U\rtimes R\in {\mathsf{Alg}}{\mathsf{Grp}}\) linear with \(U\) unipotent and \(R\) reductive is internally graded if there is a 1-parameter subgroup \(\lambda: k^{\times}\to Z(R)\) such that the adjoint action of \(\lambda(k^{\times})\curvearrowright\operatorname{Lie}U\) (the Lie algebra) has strictly positive weights.
Of interest: the \begin{align*}\[hyperbolicity conjecture\end{align*} ]. Call a projective variety over \({\mathbb{C}}\) Brody hyperbolic if any entire holomorphic map \({\mathbb{C}}\to X\) is constant.
\(\widehat{U}\) theorem can be used in situations addressed by classical GIT, e.g. curves, vector bundles or sheaves, Higgs bundles, quiver reps, etc. There is a notion of semistability in classical situations, and this allows defining moduli for unstable things. Really gives a moduli space parameterizing “stable” objects of a fixed instability type. Gives a stratification by instability types.
Prismatic cohomology: a \(p{\hbox{-}}\)adic analog of crystalline cohomology
Carries a Frobenius action.
\(H^i_{\prism}(\mathfrak{X}_{/ { \mathfrak{S} }} )\) is finitely generated over \(\mathfrak{S} = W { \left[\left[ {u} \right] \right] }\), some Witt ring?
\(\phi_{\prism}\) is a semilinear operator.
Any torsion must be \(p{\hbox{-}}\)power torsion, i.e. \(H^i_{\prism}({\mathfrak{X}}_{/ { {\mathfrak{S}}}} )_{{\operatorname{tors}}} = H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[p^{\infty}]\).
The pathological bits in all integral \(p{\hbox{-}}\)adic Hodge theories come from \(H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[u^{\infty}]\).
To study finite flat \(p{\hbox{-}}\)power group schemes, study their Dieudonne modules
Idk I just like this:
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Refs: ?
Lev Tovstopyat-Nelip, “Floer Homology and Quasipositive Surfaces,” MSU.
\begin{align*}\[contact structure\end{align*} ] on an oriented \begin{align*}\[3-manifold\end{align*} ] \(Y\): a maximally nonintegrable 2-place field \(\xi\) where \(\xi = \ker( \alpha)\) for some \(\alpha\in \Omega^1(Y)\) with \(\alpha \vee d\alpha > 0\).
A transverse knot is a knot positively transverse to \({ \left.{{ \alpha }} \right|_{{K}} } > 0\).
A knot \(K \subseteq (Y, \xi)\) is \begin{align*}\[Legendrian\end{align*} ] if \({\mathbf{T}}K\leq { \left.{{\xi}} \right|_{{K}} }\), so \({ \left.{{\alpha}} \right|_{{K}} } = 0\).
A disk \({\mathbb{D}}^2 \subseteq (Y, \xi)\) is \begin{align*}\[overtwisted\end{align*} ] if \({{\partial}}{\mathbb{D}}^2\) is Legendrian, i.e. \({ \left.{{{\mathbf{T}}{\mathbb{D}}^2}} \right|_{{{{\partial}}{\mathbb{D}}^2}} } = { \left.{{\xi }} \right|_{{{\mathbb{D}}^2}} }\).
Eliashberg: overtwisted contact structures can be studied using algebraic topology (every homotopy class of plane fields contains an overtwisted contact structures)
Tight contact structures are the interesting ones!
Define self-linking number \(\sl(\widehat{B}) = w(B) - n\).
A type of \begin{align*}\[adjunction inequality\end{align*} ] - If \(K\) is transverse in \((S^3, \xi_{\text{std}})\) then \(\sl(K) \leq 2g(K) - 1\).
For \(\Sigma\) an oriented surface with connected \(\phi\in {\operatorname{MCG}}(\Sigma, {{\partial}}\Sigma)\), define \(Y_{\phi} \mathrel{\vcenter{:}}= S\times I / (x,1) \sim (\phi(x), 0)\).
Yields \((\Sigma, \phi)\) an open book decomposition.
There is a correspondence between open book decompositions on \(Y\) and contact structures on \(Y\).
Let \(\Sigma \hookrightarrow Y\) be a Seifert surface, then it is quasipositive with respect to \(\xi\) if there exists an o.b.d. \((S, \phi)\) such that \(\Sigma \subseteq S\) is \(\pi_1{\hbox{-}}\)injective.
Lyon: every Seifert surface in a closed oriented 3-manifold is quasipositive with respect to some contact structure \(\xi\).
Measure how far a knot is from being \begin{align*}\[fibred\end{align*} ]: fibre depth.
If \(K\) is semi-quasipositive with respect to \(\xi_\text{std}\), then \(\mkern 1.5mu\overline{\mkern-1.5mu\sl\mkern-1.5mu}\mkern 1.5mu(K) = 2g(K) - 1\) Interesting #open_question: does the converge hold? I.e. if \(\mkern 1.5mu\overline{\mkern-1.5mu\sl\mkern-1.5mu}\mkern 1.5mu(K) = 2g(K) - 1\), is \(K\) semi-quasipositive?
\(\widehat{\operatorname{HFK}}\) detects genus in the sense that \(g(K)\) is the maximal nonvanishing \(\widehat{\operatorname{HFK}}(-S^3, K, i)\).
See fiberedness detection and sutured knot Floer homology.
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Refs: ?
https://arxiv.org/pdf/1904.06756.pdf
Some notes on \begin{align*}\[quadratic differentials\end{align*} ]:
Moduli space of \begin{align*}\[abelian differentials\end{align*} ] on a curve may be isomorphic to the moduli space f stability structures on the Fukaya category of the curve.
These moduli spaces admit good “wall and chamber” decompositions, with \begin{align*}\[wall crossing\end{align*} ] formulas due to Kontsevich.
Important theorems: vanishing of cohomology for \begin{align*}\[line bundles\end{align*} ] and existence of meromorphic sections:
A \begin{align*}\[principal divisor\end{align*} ] is a divisor of a meromorphic function. Taking \(\operatorname{Div}(X) / \mathop{\mathrm{Prin}}\operatorname{Div}(X)\) yields \({ \operatorname{Cl}} (X)\) the \begin{align*}\[divisor class group\end{align*} ] of \(X\).
There is a map \(\operatorname{Div}: {\operatorname{Pic}}(X) \to { \operatorname{Cl}} (X)\) sending a line bundle to its divisor class. This is an iso!
A meromorphic function has the same number of zeros and poles, i.e. \(\deg D = 0\) for \(D\in \mathop{\mathrm{Prin}}\operatorname{Div}(X)\), so degrees are well-defined for \({ \operatorname{Cl}} (X)\).
Tags: #quick_notes
Refs: ?
Kristin DeVleming, UGA AG seminar talk on moduli of quartic \begin{align*}\[K3 surfaces\end{align*} ].
Jiuya Wang’s, UGA NT seminar talk
Tags: #quick_notes
Refs: \begin{align*}\[Advice\end{align*} ]
See Manin’s universal \begin{align*}\[quantum groups\end{align*} ].
Manin defines a universal bialgebra for \(A\), which coacts on \(A\) in a universal way.
See \begin{align*}\[koszul dual\end{align*} ]
Forgetful functor from Hopf algebras to bialgebras has a left adjoint: the \begin{align*}\[Hopf envelope\end{align*} ].
Universal quantum group: take \begin{align*}\[Hopf envelope\end{align*} ] of universal bialgebra.
See \begin{align*}\[quadratic algebra\end{align*} ]
Twisting conditions for bialgebras: \(B\) is \({\mathbb{Z}}{\hbox{-}}\)graded and \(\Delta(B_n) \subseteq B\tensorpower{2}\).
Zhang twist: supplies a twisted multiplication.
Possibly related to \begin{align*}\[alpha twisted vector space\end{align*} ]?
\(\grmod{A} { { \, \xrightarrow{\sim}\, }}\grmod{A^{\phi}}\) for \(A^{\phi}\) a Zhang twist.
Morita-Takeuchi equivalence: equivalence of categories of comodules.
This talk compares cocycle twists to Zhang twists.
For \({\mathcal{O}}(G)\) the coordinate ring of \(G\in{\mathsf{Alg}}{\mathsf{Grp}}\), elements \(g\in G\) induce automorphism \(r_g, \ell_g: {\mathcal{O}}(G){\circlearrowleft}\) by left/right translation, and every twisting pair is of the form \((r_g, \ell_g^{-1})\).
Sovereign: equivalence between left and right duality functors.
Pointed algebra: simple comodules are 1-dimensional
Smash product of Hopf algebras: \(H_1\otimes H_2\) as a vector space, with a deformed multiplication.
See \begin{align*}\[quantum Yang Baxter\end{align*} ] equations. Solutions are \(R\in \mathop{\mathrm{End}}_k(V{ {}^{ \scriptstyle\otimes_{2}^{)} } }\) satisfying a tensor formula corresponding to moving strands in a braid.
Can be obtained from any braiding on \({{H}{\hbox{-}}\mathsf{coMod}}\).
Use equivalence of braided monoidal cats to get new solutions: \(\cmods{H} { { \, \xrightarrow{\sim}\, }}{{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]}{\hbox{-}}\mathsf{coMod}}{ { \, \xrightarrow{\sim}\, }}{{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]^{\sigma}}{\hbox{-}}\mathsf{coMod}}\).
Tags: #quick_notes
Refs: ?
A lead on how to formulate de Rham with a non-flat vector bundle c/o Arun: https://arxiv.org/abs/2006.02922
Hopfological algebra is related to \(d^p = 0\)?
\begin{align*}\[mock modular forms\end{align*} ], \begin{align*}\[moonshine\end{align*} ], \begin{align*}\[elliptic cohomology\end{align*} ]
Tags: #knots #concordance #geometric_topology
\(K_1, K_2\) are smoothly concordant iff there exists a smoothly embedded cylinder \(S^1\times I \hookrightarrow S^3\times I\) with \({{\partial}}(S^1\times I) = K_1 {\textstyle\coprod}-K_2\). The concordance group \(C\) is the abelian group given by knots \(K \hookrightarrow S^3\) under connect sum, modulo concordance.
If \(K_i \hookrightarrow Y_i \in \mathbb{Z}\operatorname{HS}^3\), then the \(K_i\) are homologically concordant if there is smoothly embedded cylinder \(S^1\times I \hookrightarrow W\) with \({{\partial}}(W, S^1\times I) = (Y_1, K_1) {\textstyle\coprod}(Y_2, K_2)\) with \(W\) a homology cobordism:
This yields a homological concordance group \(\widehat{C}_{\mathbb{Z}}\).
There is an injection (?) \(C_{\mathbb{Z}}\hookrightarrow\widehat{C}_{\mathbb{Z}}\) which is known by Levine not to be surjective. What can be said about the cokernel?
See \begin{align*}\[Seifert fibered space\end{align*} ], \begin{align*}\[ZHS3\end{align*} ]. These are all \begin{align*}\[homology cobordant\end{align*} ] to \(S^3\).
Proof uses \begin{align*}\[CFK\end{align*} ], a \({\mathbb{F}}[u, v]{\hbox{-}}\)module.
A knot-like complex over \(R\) is a complex \(C \in {\mathsf{gr}\,}_{{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }} \mathsf{Ch}(R)\) such that
Some examples: the knot Floer complex \begin{align*}\[CFK\end{align*} ] over a knot, \(\CFK_{{\mathbb{F}}[u, v]}(K)\). Theorem: every such complex is locally equivalent to a unique standard complex. Concordant knots produce locally equivalent complexes \(\CFK_R(K)\) for \(R \mathrel{\vcenter{:}}={\mathbb{F}}[u] \otimes_{\mathbb{F}}{\mathbb{F}}[z] / \left\langle{uv}\right\rangle\).
Set \(\mathsf{C} \mathrel{\vcenter{:}}={\operatorname{Emb}}(S^1, S^3)\), add the monoidal structure \({\sharp}\) for connect sum. Take “isotopy” category instead of homotopy category? The unit is \(\one = U\), the unknot up to isotopy. What is the stabilization of \({-}{\sharp}X\) for fixed choices of \(X\)? Or of other interesting functors? #idle_thoughts
See \begin{align*}\[torus knot\end{align*} ]
Can do base changes \(\CFK_{{\mathbb{F}}[u, v]}(M_n, K_n) \otimes_{{\mathbb{F}}[u, v]} \mathcal{X} \leadsto \CFK_{\mathcal{X}}(M_n, K_n)\) (may also need to change basis to get standard complex).
Tags: #quick_notes
\begin{align*}\[monad\|comonads\end{align*} ] in \(\mathsf{C}\): \begin{align*}\[coalgebra object\end{align*} ] in \([\mathsf{C}, \mathsf{C}]\).
\begin{align*}\[Comodules\end{align*} ] over a comonad \(T\): an object \(X\), a map \(a^\sharp: X\to TX\), and some coherence conditions. Often called \(T{\hbox{-}}\)algebras, called the category of \(T{\hbox{-}}\)comodules \({{T}{\hbox{-}}\mathsf{coMod}}(\mathsf{C})\).
A fun but non-obvious consequence of https://stacks.math.columbia.edu/tag/06WS: for \(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}\) faithfully flat, there is an equivalence of categories \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G) { { \, \xrightarrow{\sim}\, }}{\mathsf{Rep}}(G) ,\end{align*} the category of regular \(G{\hbox{-}}\)representations, i.e. \({{\Gamma(G)}{\hbox{-}}\mathsf{coMod}}\). See \begin{align*}\[regular representation\end{align*} ].
Why this is true:
\begin{align*}\[Barr-Beck\end{align*} ] says \(\tilde L\) is an equivalence under suitable conditions (\(L, R\) \begin{align*}\[conservative\end{align*} ] with \(L\) preserving \begin{align*}\[equalizers\end{align*} ]).
Set up the \begin{align*}\[adjunction\end{align*} ] \begin{align*} \mathsf{D} \mathrel{\vcenter{:}}=\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \mathrel{\vcenter{:}}=\mathsf{C} .\end{align*} Then \(LR \mathrel{\vcenter{:}}= p^*p_*\), and Barr-Beck yields \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G)\underset{\tilde{p^*}}{{ { \, \xrightarrow{\sim}\, }}} {{(p^*p_*)}{\hbox{-}}\mathsf{coMod}}({\mathsf{QCoh}}(\operatorname{Spec}R)) .\end{align*}
Use that if \(G\in{\mathsf{Aff}}{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}\) then \(\Gamma(G) \in \mathsf{Hopf}{\mathsf{Alg}_{/R}}\). Set \(\mathsf{C} \mathrel{\vcenter{:}}={\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), and \(F\in [\mathsf{C}, \mathsf{C}]\) to be \(F({-}) \mathrel{\vcenter{:}}=({-})\otimes_R \Gamma(G)\). Then there is an \begin{align*}\[equivalence of categories\end{align*} ] \begin{align*} {{F}{\hbox{-}}\mathsf{coMod}}(\mathsf{C}) { { \, \xrightarrow{\sim}\, }}{\mathsf{Rep}}(G) .\end{align*}
Then show that \(F\) is equivalent to \(p^*p_*\).
http://individual.utoronto.ca/groechenig/stacks.pdf #references
Refs: \begin{align*}\[stack\|stacks\end{align*} ] \begin{align*}\[vector bundles\|vector bundle\end{align*} ] \begin{align*}\[descent data\end{align*} ]
There is an equivalence of categories \({\mathsf{{\mathbb{R}}}{\hbox{-}}\mathsf{Mod}} { { \, \xrightarrow{\sim}\, }}\mathsf{Tw}{\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) where the latter consists of objects which are pairs \((V, f:V\to V)\) where \(f(\lambda \mathbf{v}) = \mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu \mathbf{v}\) is a structure map and \(f^2 = \operatorname{id}_V\) and morphisms \(\phi:V\to W\) that commute with the structure maps.
For field extensions \(L_{/ {k}}\), the ring morphism \(k\hookrightarrow L\) yields \(\operatorname{Spec}L \to \operatorname{Spec}k\), which behaves like a \begin{align*}\[covering space\end{align*} ] with \(\mathop{\mathrm{Deck}}(\operatorname{Spec}L _{/ {\operatorname{Spec}k}} ) \cong { \mathsf{Gal}} (L_{/ {k}} )\).
Vector bundles on \(\operatorname{Spec}k\) correspond to \({\mathsf{k}{\hbox{-}}\mathsf{Mod}}\), and Galois-equivariant vector bundles on \(\operatorname{Spec}L\) will correspond to vector bundles on the quotient \(\operatorname{Spec}k\).
\(R\in {\mathsf{Alg}}_{/ {A}}\): a ring morphism \(A\to R\).
Given \(f\in {\mathbb{Z}}[x_1,\cdots, x_n]\), taking the zero locus in a ring \(R\) yields a functor \(\mathsf{CRing}\to {\mathsf{Set}}\). To do this with \(f\in A[x_1,\cdots, x_n]\) for \(A\in \mathsf{CRing}\), one needs \(R\in {\mathsf{Alg}}_{/ {A}}\), so this yields a functor \({\mathsf{Alg}}_{/ {A}} \to {\mathsf{Set}}\).
Think of spaces as functors \(X\in [\mathsf{CRing}, {\mathsf{Set}}]\), then \(\operatorname{Spec}R \mathrel{\vcenter{:}}=\mathsf{CRing}(R, {-})\), so \(R\) corepresents \(\operatorname{Spec}R\) in \(\mathsf{CRing}\).
Can represent \(R \left[ { \scriptstyle { {f}^{-1}} } \right] = R[t]/\left\langle{tf-1}\right\rangle\).
Standard open subfunctors: \(\operatorname{Spec}R \left[ { \scriptstyle { {f_i}^{-1}} } \right] \to \operatorname{Spec}R\). These form an open cover if \(\left\langle{f_i}\right\rangle = \left\langle{1}\right\rangle\).
If \(k\in \mathsf{Field}\), there is an equivalence \(\operatorname{Spec}R(k) \cong Z_f(k)\), the zeros of \(f\) in \(k\). Then \(\operatorname{Spec}R \left[ { \scriptstyle { {h}^{-1}} } \right](k) = Z_f(k)\setminus Z_h(k)\) for \(R = {\mathbb{Z}}[x_1,\cdots, x_n]/\left\langle{f}\right\rangle\).
Analog of 2-dimensional \({\mathbb{C}}{\hbox{-}}\)module over a ringer ring: the free \(R{\hbox{-}}\)module \(R{ {}^{ \scriptscriptstyle\times^{2} } }\) of rank 2.
\({\mathbb{P}}^1_{{\mathbb{Z}}}: \mathsf{CRing}\to{\mathsf{Set}}\) is the functor sending \(R\) to the set of direct summands \(M \leq R{ {}^{ \scriptscriptstyle\times^{2} } }\) for which there’s an open covering corresponding to \(\left\{{h_i}\right\}\) where \(M \left[ { \scriptstyle { {h_i}^{-1}} } \right] = M\otimes_R R \left[ { \scriptstyle { {h_i}^{-1}} } \right]\) is a free \(R{\hbox{-}}\)module of rank 1 for all \(i\).
For \(S\in{\mathsf{Alg}_{/R}}\), we have \(\alpha: R\to S\) and for \(N\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}}\) we can forget the module structure along this map by defining \begin{align*} R\times N &\to N \\ (r, n) &\mapsto \alpha(r) \cdot n .\end{align*} This induces a \begin{align*}\[restriction functor\end{align*} ] \(\operatorname{res}_{\alpha}: {\mathsf{S}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).
Conversely we can tensor \(R{\hbox{-}}\)modules up to \(S{\hbox{-}}\)modules to get a functor \(S\otimes_R({-})\), where the interesting bit is \(s\otimes(rm) \mathrel{\vcenter{:}}=\alpha(r) (s\otimes m) = (\alpha(r)s)\otimes m\).
This yields an adjunction: \begin{align*} \adjunction{({-})\otimes_R S}{\operatorname{res}_{\alpha}}{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}} .\end{align*}
Any reasonable property of modules should be preserved by base change!
Descent for modules: when does \(M\otimes_R S\) having property \(P\) as an \(S{\hbox{-}}\)module descend to \(M\) having property \(P\) has an \(R{\hbox{-}}\)module?
Left adjoints are right exact (LARE). In particular, \begin{align*}\[base change\end{align*} ] is right exact, but not always left exact: take \(\alpha: {\mathbb{Z}}\to {\mathbb{Z}}/2\), take the SES \(0 \to {\mathbb{Z}}\xrightarrow{2} {\mathbb{Z}}\to {\mathbb{Z}}/2\to 0\), and tensor with \({\mathbb{Z}}/2\). So an \(R{\hbox{-}}\)algebra \(S\) is flat precisely when the base change \(S\otimes_R({-})\) is exact.
\(S\) is \begin{align*}\[faithfully flat\end{align*} ] when \(S\otimes_R M = 0\implies M=0\). Allows checking things after base-changing to \(S\): - Exactness of any sequence, so in particular injectivity/surjectivity - Finite generation (over \(R\) vs \(S\)) - Projectivity, - Flatness - If \(R\to S\) is faithfully flat and \(R\to T\) is an arbitrary ring morphism, the co-base change \(T\to S\otimes_R T\) is faithfully flat.
General idea: \(R{\hbox{-}}\)modules \(M\) can be specified by \(S\otimes_R M\) along with \begin{align*}\[descent data\end{align*} ].
\begin{align*}\[Faithfully flat descent\end{align*} ] : there is an equivalence of categories \({\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)\),
Given \(F\in [\mathsf{A}, \mathsf{B}]\) and \(G\in [\mathsf{A}, \mathsf{C}]\), the left \begin{align*}\[Kan extension\end{align*} ] of \(G\) along \(F\) is a functor \(L\in [B, C]\) and a sufficiently universal natural transformation \(\alpha\in [G, LF]\).
Tags: #quick_notes
Tags: #terms_and_questions
Tags: #quick_notes
Tags: #reading_notes #derived #infinity_cats
Derived AG: https://people.math.harvard.edu/~lurie/papers/DAG-X.pdf
\begin{align*}\[dg Lie algebras\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-03-49.png\end{align*} ]
\begin{align*}\[elliptic curve\|elliptic curve\end{align*} ] and \begin{align*}\[deformation theory\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-05-28.png\end{align*} ]
\begin{align*}\[presentable infinity category\end{align*} ]. \begin{align*}\[deformation-obstruction theory\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-08-54.png\end{align*} ]
\begin{align*}\[k-linear category\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_00-19-40.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-21-36.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-28-30.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-30-48.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-33-46.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_00-34-14.png\end{align*} ]
Weak weak approximation would imply a positive answer to the \begin{align*}\[inverse Galois problem\end{align*} ].
\begin{align*}\[attachments/2021-10-05_20-02-50.png\end{align*} ]
Tags: #stable_homotopy #physics #summaries
Refs: \begin{align*}\[Elliptic cohomology\end{align*} ], \begin{align*}\[Thom-Dold\end{align*} ], \begin{align*}\[Orientability of spectra\|orientability\end{align*} ], \begin{align*}\[formal group law\end{align*} ], \begin{align*}\[ring spectra\end{align*} ], \begin{align*}\[Bousfield localization\end{align*} ], \begin{align*}\[Topological modular forms\|tmf\end{align*} ],
Reference: M-theory, type IIA superstrings, and elliptic cohomology https://arxiv.org/pdf/hep-th/0404013.pdf
\begin{align*}\[attachments/2021-10-05_20-39-39.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-40-20.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-41-16.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-41-33.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-41-56.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-42-42.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-43-37.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-44-09.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-44-36.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-45-25.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-46-47.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-48-43.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-51-54.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_20-51-38.png\end{align*} ]
Hom is a \begin{align*}\[continuous functor\end{align*} ], i.e. it preserves limits in both variables. Just remember that the first argument is contravariant, so \begin{align*} \cocolim_i \cocolim_j \mathsf{C}(A_i, B_j) = \mathsf{C}(\colim_i A_i, \cocolim_j B_j) .\end{align*}
\begin{align*}\[tannaka duality\end{align*} ] and \begin{align*}\[tannaka reconstruction\end{align*} ] :
\begin{align*}\[attachments/2021-10-05_23-01-03.png\end{align*} ]
\begin{align*}\[attachments/2021-10-05_23-04-52.png\end{align*} ]
How \begin{align*}\[K-Theory\end{align*} ] goes:
Minor aside: \({\mathbf{B}}\mathsf{C} \mathrel{\vcenter{:}}={ {\left\lvert {{ \mathcal{N}({\mathsf{C}}) }} \right\rvert} }\).
Start with the category of \begin{align*}\[elliptic curve\|elliptic curves\end{align*} ] : should be pointed \begin{align*}\[algebraic group\end{align*} ], so a \begin{align*}\[slice category\|coslice category\end{align*} ] over a terminal object..?
Refs: \begin{align*}\[algebra valued differential forms\end{align*} ]
https://www.wikiwand.com/en/Lie_algebra-valued_differential_form
\begin{align*}\[Siegel modular forms\end{align*} ] arise as vector-valued differential forms on Siegel modular varieties? See the following paper for leads: https://arxiv.org/abs/math/0605346
\begin{align*}\[adS correspondence conjecture\end{align*} ] :
Tags: #stable_homotopy
Producing a LES:
Integration pairing: for \(E \in {\mathsf{SHC}}(\mathsf{Ring})\), \begin{align*} E^*X &\longrightarrow E_* X \\ \omega \in [\mathop{\mathrm{{\Sigma_+^\infty}}}X, E] &\longrightarrow\alpha \in [{\mathbb{S}}, E\wedge X] \\ \\ {\mathbb{S}}\xrightarrow{\alpha} E \wedge X \cong E\wedge{\mathbb{S}}\wedge X &\cong E \wedge\mathop{\mathrm{{\Sigma_+^\infty}}}X \xrightarrow{1\wedge\omega } E{ {}^{ \scriptscriptstyle\wedge^{2} } } \xrightarrow{\mu} E .\end{align*}
Tags: #category_theory #simplicial #infinity_cats
Recall \({\mathsf{sSet}}= [\Delta^{\operatorname{op}}, {\mathsf{Set}}] = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{Set}}) = {\mathsf{Set}}^{\Delta^{\operatorname{op}}}\).
For \(x_0 \in \mathsf{C}\), a cone from \(x_0\) to \(F\in [J, C]\) for \(J\) any diagram category is a family \(\psi_x: x_0 \to F(x)\) making diagrams commute:
Free \begin{align*}\[cocompletion\end{align*} ] of a category: \(\mathsf{C} \mapsto [\mathsf{C}, {\mathsf{Set}}]\).
Cauchy completeness for a category: closure under all \begin{align*}\[colimits\end{align*} ] that are preserved by every functor.
\begin{align*}\[Subfunctor\end{align*} ] : \(G\leq F\) iff \(G(x) \subseteq F(x)\) and for all \(x \xrightarrow{f} y\), require \(F(f)(G(x)) \subseteq G(y)\).
References: https://arxiv.org/pdf/0801.3480.pdf and https://people.math.umass.edu/~gwilliam/thesis.pdf
Tags: #reading_notes #lie_algebras
String structures on \(X\): spin structures on \({\Omega}X\).
Defining algebra-valued forms when curvature doesn’t vanish:
See \begin{align*}\[factorization algebra\end{align*} ]
\begin{align*}\[derived infinity category\end{align*} ] : \begin{align*}\[differential graded nerve\end{align*} ] of subcategory of \begin{align*}\[fibrant objects\end{align*} ]. Always a \begin{align*}\[stable infinity category\end{align*} ], and localizes at weak equivalences.
Alternatively: take subcategory of fibrant objects, observe \begin{align*}\[enrichment\end{align*} ] over chain complexes, apply \begin{align*}\[Dold-Kan\end{align*} ] to get a simplicial enrichment, then take the \begin{align*}\[homotopy coherent nerve\end{align*} ] or \begin{align*}\[simplicial nerve\end{align*} ].
Getting a chain complex from a \begin{align*}\[simplicial set\end{align*} ] : take free \({\mathbb{Z}}{\hbox{-}}\)modules levelwise, then apply \begin{align*}\[Dold-Kan\end{align*} ].
How (I think?) Postnikov and Whitehead towers are related:
Tags: #idle_thoughts
Idk this weird thing
Tags: #terms_and_questions
Tags: #terms_and_questions
https://math.stanford.edu/~conrad/papers/hypercover.pdf
Tags: #idle_thoughts #simplicial
Not sure how to get this to work yet, but here’s the condition for a functor to be a sheaf:
But we can write \(n{\hbox{-}}\)fold intersections as \begin{align*}\[fiber products\end{align*} ] :
So the condition of \({\mathcal{F}}\) being a sheaf seems to look like letting \({\mathcal{U}}\rightrightarrows X\) be an open cover, setting \(M = {\textstyle\coprod}U_i\), then applying a \begin{align*}\[bar construction\end{align*} ] \begin{align*} M: M{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{1} } } \leftarrow M{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } } \leftarrow\cdots .\end{align*} Then apply \({\mathcal{F}}\), and look at some kind of image sequence? And ask for exactness for \(n\) many levels to get a sheaf, \begin{align*}\[stack\end{align*} ], etc:
The problem is that I don’t really know how to relate the bottom line (whose exactness is the usual condition for sheaves, stacks, etc) to the intermediate steps. This seems like it wants \({\mathcal{F}}({\textstyle\coprod}{-}) = \prod {\mathcal{F}}({-})\), so it commutes with (co?)limits, since probably contravariant functors send coproducts to products. Moreover the bar construction in the 2nd line might form a simplicial object? And the condition of satisfying \begin{align*}\[descent\end{align*} ] is maybe related to either this being a \begin{align*}\[simplicial object\end{align*} ], or its image in the bottom line assembling to a simplicial object, since there are clear degeneracy maps and one would want sections in order to build face maps. Super vague, there are a lot of details missing here!!
Tags: #category_theory
\begin{align*}\[2-category\end{align*} ] : a category \begin{align*}\[Enriched category\|enriched\end{align*} ] in small categories, so hom sets are categories and compositions form bifunctors.
\begin{align*}\[classifying space\end{align*} ] of a 2-cat: replace morphism cats \(C(x, y)\) with \({\mathbf{B}}C(x, y)\) to get a topological 1-cat, then take \({\mathbf{B}}C \mathrel{\vcenter{:}}={ {\left\lvert {{ \mathcal{N}({C}) }} \right\rvert} }\).
For \(F:\mathsf{C}\to \mathsf{D}\), fixing \(p\in D\), can form a \begin{align*}\[homotopy fiber\end{align*} ] 2-category \(y//F\).
Then \({\mathbf{B}}F: {\mathbf{B}}C\to {\mathbf{B}}D\) is a homotopy equivalence of spaces if \(B(y//F) \simeq{\operatorname{pt}}\) is contractible for all \(y\in \mathsf{D}\).
\begin{align*}\[homotopy fiber\end{align*} ] cat: \(y//F\) is a lax \begin{align*}\[comma category\end{align*} ].
\begin{align*}\[lax functor\end{align*} ] :
Tags: #representation_theory #terms_and_questions #number_theory #langlands
\begin{align*}\[Cartan\end{align*} ] subgroup
\begin{align*}\[Borel\end{align*} ] subgroup
\begin{align*}\[Parabolic\end{align*} ] subgroup
\begin{align*}\[local fields\end{align*} ]
\begin{align*}\[valuation\end{align*} ] : For \(v: k \to G\cup\left\{{\infty}\right\}\) and \(G\in {\mathsf{Ab}}\) \begin{align*}\[totally ordered\end{align*} ].
\begin{align*}\[global field\end{align*} ] : algebraic number fields, function fields of \begin{align*}\[algebraic curve\|algebraic curves\end{align*} ] over finite fields (so finite extensions of \({\mathbb{F}}_q { \left( {(} \right) } t))\).
Note the closed point of \(\operatorname{Spec}{ {\mathbb{Z}}_p }\) is \({\mathbb{F}}_p\) and the generic point is \({ {\mathbb{Q}}_p }\).
\begin{align*}\[nonarchimedean field\end{align*} ]
\begin{align*}\[proper morphism\end{align*} ]
\begin{align*}\[Iwahori\end{align*} ] subgroup
Fun fact: \(p{\hbox{-}}\)torsion in an \begin{align*}\[ideal class group\end{align*} ] was the main obstruction to a direct proof of FLT. Observed by Kummer.
Main conjecture of \begin{align*}\[Iwasawa theory\end{align*} ] : two methods of defining \(p{\hbox{-}}\)adic \(L{\hbox{-}}\)functions should coincide. Proved by Mazur/Wiles for \({\mathbb{Q}}\), all totally real number fields by Wiles.
Actual definition of Dirichlet characters:
Define: \begin{align*}\[geometric fiber\end{align*} ]
Reductive, semisimple, simply connected, etc for \(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {S}}\): affine and smooth over \(S\), where geometric fibers are reductive. s.s., etc algebraic groups.
\begin{align*}\[etale morphism\end{align*} ]
Central extension
Fiber functor
Algebraic fundamental group
Certain groups that become isomorphic after field extensions have related automorphic representations.
Langlands dual: \({\mathcal{L}}(G)\) controls \({\mathsf{G}{\hbox{-}}\mathsf{Mod}}\) somehow, arises as an extension \({ \mathsf{Gal}} (k^s _{/ {k}} ) \to {\mathcal{L}}(G) \to H\) where \(H \in \operatorname{Lie}{\mathsf{Grp}}_{/ {{\mathbb{C}}}}\).
A connected reductive algebraic group over a separably closed field \(k\) is uniquely determined by its root datum.
\begin{align*}\[Langlands dual\end{align*} ] : take root datum, dualize datum, take associated group.
Langlands’ strategy for proving local and global conjectures: \begin{align*}\[Arthur-Selberg trace formula\end{align*} ].
Equivalence of orbital integrals can somehow be related to \begin{align*}\[Springer Fiber\end{align*} ]??
Starting point for Langlands: \begin{align*}\[Artin reciprocity\end{align*} ], generalizing \begin{align*}\[quadratic reciprocity\end{align*} ].
\begin{align*}\[Chebotarev density\end{align*} ] theorem is a generalization of \begin{align*}\[Dirichlet's theorem on arithmetic progressions\end{align*} ].
“The Langlands conjectures associate an automorphic representation of the adelic group \(\operatorname{GL}_n({\mathbb{A}}_{/ {{\mathbb{Q}}}} )\) to every \(n{\hbox{-}}\)dimensional irreducible representation of the Galois group, which is a \begin{align*}\[cuspidal representation\end{align*} ] if the Galois representation is irreducible, such that the \begin{align*}\[Artin L function\end{align*} ] of the Galois representation is the same as the \begin{align*}\[automorphic L function\end{align*} ] of the automorphic representation”
Serre’s modularity conjecture: an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form
\({\mathbb{A}}_{/ {{\mathbb{Q}}}}\): keeps track of all of the completions of \({\mathbb{Q}}\) simultaneously.
Reciprocity conjecture: a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an \(L\)-group
\begin{align*}\[geometric Langlands\end{align*} ] : relates \begin{align*}[l-adic representations](#l-adic-representations-1)\end{align*} of the \begin{align*}\[étale fundamental group\end{align*} ] of an \begin{align*}\[algebraic curve\end{align*} ] to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.
2018: Lafforgue established \begin{align*}\[Global Langlands correspondence\|global Langlands\end{align*} ] for automorphic forms to \begin{align*}\[Galois representations\end{align*} ] for connected reductive groups over global function fields
\begin{align*}\[purity\end{align*} ] : happens in a specific codimension
Tags: #homological_stability #representation_theory
Reference: Church-Ellenberg-Farb
Smallest subrings of \begin{align*}\[Chow ring\|Chow\end{align*} ] closed under \begin{align*}\[pushforward\end{align*} ] by forgetful/gluing maps between various \({ \mathcal{M}_{g, n} }\).
Can push through the \begin{align*}\[cycle class map\end{align*} ], unknown these are isomorphic.
Can get a surjection \({\mathbb{Q}}[\kappa_i] \to H^*({ \mathcal{M}_{g, n} })\) for degree high enough.
Main result: dimensions of representation stabilize.
Sequence of \(S_n{\hbox{-}}\)reps converted into a single \({\mathsf{FI}}{\hbox{-}}\)module.
\({\mathsf{{\mathsf{FI}}}{\hbox{-}}\mathsf{Mod}} \mathrel{\vcenter{:}}= F\in {\mathsf{Fun}}({\mathsf{FI}}, {\mathsf{k}{\hbox{-}}\mathsf{Mod}})\).
Gradings: functors from \({\mathbb{N}}\to {\mathsf{k}{\hbox{-}}\mathsf{Mod}}\).
Tags: #modular_forms #moduli_spaces #stacks
Tags: #reading_notes Refs: \begin{align*}\[modular form\end{align*} ]
Reference: see https://www.math.purdue.edu/~arapura/preprints/shimura2.pdf
\({\operatorname{SL}}_2({\mathbb{R}})\curvearrowright{\mathbb{H}}\) transitively by linear fractional transformations, and \({\operatorname{Stab}}(i) = {\operatorname{SO}}(2)\). Thus one can realize \({\mathbb{H}}\cong {\operatorname{SL}}_2({\mathbb{R}})/{\operatorname{SO}}_2({\mathbb{R}})\).
Applying a homothety to a \begin{align*}\[lattice\end{align*} ] \(\Lambda\) yields \(L_\tau \mathrel{\vcenter{:}}={\mathbb{Z}}+ {\mathbb{Z}}\tau\) for some \(\tau\in{\mathbb{H}}\) and \(\Lambda \cong L_\tau\). Writing an elliptic curve as \({\mathbb{C}}/L_\tau\), the moduli of elliptic curves is given by \begin{align*} A_1\mathrel{\vcenter{:}}={\operatorname{SL}}_2({\mathbb{Z}})\diagdown{\mathbb{H}}\cong \dcoset{{\operatorname{SL}}_2({\mathbb{R}})}{{\operatorname{SL}}_2({\mathbb{Z}})}{{\operatorname{SO}}_2({\mathbb{R}})} .\end{align*} This quotient is Hausdorff, and \(A_1 \xrightarrow{\sim} {\mathbb{C}}\) as topological spaces. Somehow this comes from “gluing the two bounding lines of \(F\) and folding the circular boundary in half,” yielding the sphere minus a point.
\(-I\) acts trivially on \({\mathbb{H}}\), so this factors through \(\Gamma \mathrel{\vcenter{:}}={\operatorname{PSL}}_2({\mathbb{Z}}) \mathrel{\vcenter{:}}={\operatorname{SL}}_2({\mathbb{Z}})/\left\langle{\pm I}\right\rangle\).
Letting \(S= (z\mapsto -1/z) = \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right), T = (z\mapsto z+1) = \left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)\), there are fundamental domains:
\(i\) has \begin{align*}\[isotropy\end{align*} ] \(\left\langle{S}\right\rangle\), \(\zeta_3\) has \(\left\langle{ST}\right\rangle\), and \(\zeta_3^2\) has \(\left\langle{TS}\right\rangle\). Applying \(S\) and \(T^{\pm 1}\) to the fundamental domain \(F\) tiles \({\mathbb{H}}\) by hyperbolic triangles.
\({\operatorname{PSL}}_2({\mathbb{Z}}) = \left\langle{S, T}\right\rangle\).
Maps \(f: A_1\to {\mathbb{C}}\) are continuous iff their pullbacks along \(\pi: {\mathbb{H}}\to A^1\) are continuous, so these are necessarily \(\Gamma{\hbox{-}}\)invariant functions.
\(f\) is \begin{align*}\[automorphic\end{align*} ] with automorphy factor \(\phi_\gamma(z)\) iff \begin{align*} f(\gamma z) = \phi_\gamma(z) f(z) .\end{align*} For any two such functions, their ratio \(g=f_1/f_2\) satisfies \(g(\gamma z) = g(z)\).
\(f\) is weakly modular of weight \(2k\) if \begin{align*} f(z) = (cz+d)^{-2k} f(\gamma z), \gamma \mathrel{\vcenter{:}}={ \begin{bmatrix} {a} & {b} \\ {c} & {d} \end{bmatrix} } .\end{align*}
Note that \begin{align*} {\frac{\partial }{\partial z}\,} {az+b \over cz +d }= {ad-bc \over (cz+d)^2} = {1\over (cz+d)^2} ,\end{align*} and so a meromorphic form \(\omega = f(z) \,dz\) transforms under \(\gamma\) as \begin{align*} \gamma \cdot f(z)\,dz= f(\gamma \cdot z) d(\gamma\cdot z) = (cz+d)^{-2}f(\gamma\cdot z)\,dz .\end{align*}
So weakly modular forms of weight \(2k\) are those form which \(\omega^{\otimes k}\) is invariant.
Dropping the weakly adjective involves imposing holomorphy conditions at \(\infty\). \(f\) is a (standard) \begin{align*}\[modular form\end{align*} ] of \begin{align*}\[weight of a modular form\|weight\end{align*} ] \(2k\) if \(f\) is weakly modular, holomorphic on \({\mathbb{H}}\), and the Fourier coefficients satisfy \(a_{<0} = 0\).
Poisson summation: if \(f:{\mathbb{R}}\to {\mathbb{C}}\) is a \begin{align*}\[Schwartz function\end{align*} ] (smooth and super-polynomial decay), then \begin{align*} \sum_{n \in \mathbb{Z}} f(n)=\sum_{n \in \mathbb{Z}} \widehat{f}(n) .\end{align*}
\(\zeta(s)\) is the \(L{\hbox{-}}\)function associated to the trivial Galois representation \begin{align*} \rho_{\mathop{\mathrm{Triv}}}: G_{\mathbb{Q}}\to {\mathbb{C}}^{\times} .\end{align*} \(L{\hbox{-}}\)functions coming from arbitrary 1-dim reps will correspond to Dirichlet characters by Kronecker-Weber, and are referred to as Dirichlet \(L{\hbox{-}}\)functions.
\(\Gamma(1) \mathrel{\vcenter{:}}={\operatorname{SL}}_2({\mathbb{Z}})\), and principal congruence subgroups of level \(N\) for \(\Gamma(1)\) are defined as \begin{align*} \Gamma(N) \mathrel{\vcenter{:}}=\ker\qty{\Gamma(1) \twoheadrightarrow{\operatorname{SL}}_2({\mathbb{Z}}/N)} = \left\{{M\in \Gamma(1) {~\mathrel{\Big\vert}~}M\cong I \operatorname{mod}N}\right\} ,\end{align*} so the kernels of reduction mod \(N\). \begin{align*}\[Congruence subgroups\end{align*} ] are any subgroups \(H\) such that \(\Gamma(N) \subseteq H \leq \Gamma(1)\) for some \(N\).
Letting \(\Gamma(N)\) act on \({\mathbb{H}}\) or \({\mathbb{H}}^*\), one can define \begin{align*}\[modular curves\end{align*} ] \begin{align*} X(\Gamma) &\mathrel{\vcenter{:}}=\Gamma\diagdown {\mathbb{H}}^* \\ Y(\Gamma) &\mathrel{\vcenter{:}}=\Gamma\diagdown {\mathbb{H}} ,\end{align*} where \({\mathbb{H}}^* = {\mathbb{H}}\cup({\mathbb{Q}}\cup\left\{{\infty}\right\}) \subset {\mathbb{P}}^1({\mathbb{C}})\).
The inclusions \(\Gamma \hookrightarrow\Gamma(1)\) induce a branched cover \(X(\Gamma) \twoheadrightarrow X(\Gamma(1)) = A^1 \cong {\mathbb{P}}^1({\mathbb{C}})\).
The genera of these curves can be computed using \begin{align*}\[Riemann-Hurwitz\end{align*} ] : \begin{align*} 2 g(Y)-2=(2 g(X)-2) d+\sum_{y \in Y}\left(e_{y}-1\right) ,\end{align*} yielding for \(N\geq 3\), \(g(X(\Gamma(N)))\) is given by \begin{align*} g=1+\frac{d(N-6)}{12 N} {\quad \operatorname{where} \quad} d=\frac{1}{2}[\Gamma(1): \Gamma(N)]=\frac{N^{3}}{2} \prod\left(1-\frac{1}{p^{2}}\right) .\end{align*}
For \(X\) a smooth curve and \(D\in \operatorname{Div}(X)\), set \begin{align*} \Omega^1_X(D) \mathrel{\vcenter{:}}=\Omega^1_X \otimes{\mathcal{O}}_X(D) \cong {\mathcal{O}}_X(\omega + D) \end{align*} where \(\omega\) is the canonical divisor. Then \({{\Gamma}\qty{X; \Omega^1_X(D)} }\) is the space of meromorphic 1-forms \(\omega\) such that \(\operatorname{Div}(\omega) + D \geq 0\) is effective.
Define \(M_{2k}(\Gamma)\) to be the space of weight \(2k\) modular forms, and \(S_{2k}\) the space of cusp forms. Then \(\bigoplus_k S_{2k} \in {\mathsf{gr}\,}^{\mathbb{Z}}{\mathsf{Alg}}_{/ {{\mathbb{C}}}} ^{\mathrm{fg}}\), and for \({\operatorname{SL}}_2({\mathbb{Z}})\) this algebra is generated by the Eisenstein series \(G_4\) and \(G_6\).
A contravariant functor \(F\) admits a \begin{align*}\[fine moduli space\end{align*} ] \({\mathbf{B}}F\) if \(F\) is representable by \({\mathbf{B}}F\), i.e. \(F({-}) \cong \mathop{\mathrm{Hom}}({-}, {\mathbf{B}}F)\). By Yoneda, \(F\) admits a universal family \({\mathbf{E}}F \to {\mathbf{B}}F\) so that \(F(X)\) is the pullback of it under some map \(X\to {\mathbf{B}}F\).
The functor \(F({-})\) sending \(X\) to isomorphism classes of elliptic curves over \(X\) admits \(Y(1)\) as a \begin{align*}\[fine moduli space\|coarse moduli space\end{align*} ] and not a fine one, since there are nontrivial families with constant \(j{\hbox{-}}\)invariant. Despite this, \(E\to j(E)\) gives a bijection between isomorphism classes of elliptic curves and points of \(Y(1)\).
A level \(N\) structure is a basis for \(H_1(E; {\mathbb{Z}}/N)\), which is symplectic since it carries a pairing with intersection matrix \({ \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} }\).
Moduli interpretations:
An \begin{align*}\[elliptic curve\end{align*} ] \(E\) over a \begin{align*}\[scheme\end{align*} ] \(S\) is a \begin{align*}\[smooth proper morphism\end{align*} ] \(f:E\to S\) with a section such that the closed fibers of \(f\) are genus 1 curves.
Look up the \begin{align*}\[Weil Pairing\end{align*} ] \(e_n\).
A level \(N\) structure is a pair of points \(P, Q \in E[N]\) generating a subgroup with that \(e_n(P, Q) = \zeta_N\) is a primitive \(N\)th root of unity. More generally, for curves over schemes, this is a pair of sections inducing level structures on closed fibers.
For \(N=2\), \(Y(2) = \operatorname{Spec}{\mathbb{Z}} { \left[ {t, {1\over t(t-1)}} \right] }\) as a coarse moduli space, and a corresponding almost-universal family \(y^2z = x(x-z)(x-tz)\).
For \(N=3\), \(Y(2) = \operatorname{Spec}R { \left[ {t, {1\over t^3-1}} \right] }\) where \(R \mathrel{\vcenter{:}}={\mathbb{Z}} { \left[ {{1\over 3}, \zeta_3^2} \right] }\) for \(\zeta_3\) a primitive third root of unity. The universal family is \(x^3 + y^3 + z^3 = 3txyz\), where the level 3 structure is given by the sections \({\left[ {-1, 0, 1} \right]}, [-1, \zeta_3^2, 0]\).
Can view an elliptic curve as a pair \((X, p)\) where \(X\) is a compact Riemann surface with \(\dim_{\mathbb{C}}H^0(X; \Omega^1_X) = 1\) and \(p\) is a point.
Can define a lattice \(\Lambda \subseteq V\) in an arbitrary vector space as a discrete \begin{align*}\[cocompact\end{align*} ] subgroup, so \(V/\Lambda\) is compact.
The order of a function \(f\) at \(x\) is given by
\begin{align*} \text { ord }_{x}(f):= \begin{cases}0 & \text { if } \mathrm{f} \text { is holomorphic and non-zero at } x \\ k & \text { if } \mathrm{f} \text { has a zero of order } k \text { at } x \\ -k & \text { if } \mathrm{f} \text { has a pole of order } k \text { at } x\end{cases} .\end{align*}
Can define divisors as maps \(D:X\to {\mathbb{Z}}\) where cofinitely many points are sent to zero. The map \({\operatorname{Ord}}_{{-}}(f)\) is a divisor associated to any function \(f\), denoted \((f)\).
Divisors \((f)\) for \(f\) meromorphic are principal, and setting \(\deg(\sum n_i p_i) \mathrel{\vcenter{:}}=\sum n_o\), it turns out that \(\deg((f)) = 0\) for \(f\) principal.
The period map is defined as \begin{align*} \Phi: H_{1}(X, \mathbb{Z}) & \rightarrow \mathbb{C} \\ \gamma & \mapsto \int_{\gamma} \omega .\end{align*} For a fixed nonzero holomorphic 1-form \(\omega\), there is a group pf \begin{align*}\[periods\end{align*} ] which forms a lattice over \({\mathbb{C}}\): \begin{align*} \Lambda \mathrel{\vcenter{:}}=\left\{{\int_\gamma \omega \in {\mathbb{C}}{~\mathrel{\Big\vert}~}\gamma \in H_1(X; {\mathbb{Z}})}\right\} = \operatorname{im}\Phi .\end{align*} One can recover \((X, P)\) as \(({\mathbb{C}}/\Lambda(\omega), 0)\)
To pick back up: https://repositorio.uniandes.edu.co/bitstream/handle/1992/43725/u830743.pdf?sequence=1
Tags: #category_theory #homotopy_theory
\begin{align*}\[mapping cone\end{align*} ] as a \begin{align*}\[pushout\end{align*} ] and \begin{align*}\[mapping fiber\end{align*} ] as \begin{align*}\[pullback\end{align*} ] :
Tags: #probability #review_material #undergraduate
Tags: #combinatorics #review_material #undergraduate
Prop: the number of \begin{align*}\[integer compositions\end{align*} ] of \(n\) into exactly \(k\) parts is \({n-1 \choose k-1}\), so \(\sum a_i = n\) with \(a_i\geq 1\) for all \(i\).
Prop: the number of \begin{align*}\[compositions\end{align*} ] of \(n\) into any number of parts is \(2^{n-1}\).
Prop: the number of weak compositions of \(n\) into exactly \(k\) parts is \({n+k-1 \choose k-1}\).
Quotient set of compositions by permutations to get \begin{align*}\[integer partitions\end{align*} ].
\begin{align*}\[Stirling numbers\end{align*} ] : the number of ways to partition an \(n\) element set into exactly \(k\) nonempty unlabeled disjoint blocks whose disjoint union is the original set.
Tags: #stable_homotopy #k_theory
\(K^\mathsf{Alg}({\mathbb{S}})\) is of interest due to Waldhausen’s stable parameterized \begin{align*}\[h-cobordism theorem\end{align*} ].
A first approximation: \({\mathsf{K}}^\mathsf{Alg}(L_{K(n)} {\mathbb{S}})\), so localize with respect to \begin{align*}\[Morava K-theory\end{align*} ]
Rognes extends \begin{align*}\[Lichtenbaum-Quillen conjectures\end{align*} ] and develops a theory for Galois extensions of \({\mathbb{S}}{\hbox{-}}\)algebras
\begin{align*}\[Redshift\end{align*} ] : \begin{align*}\[algebraic K-theory\end{align*} ] increases \begin{align*}\[chromatic complexity\end{align*} ] by one.
Refs: \begin{align*}\[A1 Homotopy\end{align*} ]
Context: the infinity category of spaces, i.e. \begin{align*}\[homotopy types\end{align*} ]
Take smooth manifolds, take the \begin{align*}\[Yoneda embedding\end{align*} ] to \(\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Mfd}})\): these satisfy a Mayer-Vietoris gluing property, and homotopy invariance in the sense that \begin{align*} \mathop{\mathrm{Hom}}({-}, X) \cong \mathop{\mathrm{Hom}}({-}\times I, X) .\end{align*}
AG setting: \(\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Sch}}_{/k})\), send to presheaves to define \begin{align*}\[motivic spaces\end{align*} ].
See \begin{align*}\[Betti realization\end{align*} ] for \(k={\mathbb{C}}\): \({\mathsf{sm}}{\mathsf{Sch}}_{/{\mathbb{C}}}\to {\mathsf{Spaces}}\) where \(X\mapsto X({\mathbb{C}})\).
From topology: identify \({\mathbf{B}}{\operatorname{U}}_n({\mathbb{C}}) = {\operatorname{Gr}}_n({\mathbb{C}})\) to get \begin{align*} { \mathsf{Vect} }_{/{\mathbb{C}}}^{\operatorname{rank}= n}(U) \cong \pi_0 \mathop{\mathrm{Maps}}(U, {\mathbf{B}}{\operatorname{U}}_n({\mathbb{C}})) .\end{align*}
Problem in AG: there are two rank 2 \begin{align*}\[vector bundles\end{align*} ] on \({\mathbb{P}}^1 \times{\mathbb{A}}^1\) whose fibers over 0 and 1 are \({\mathcal{O}}^2\) and \({\mathcal{O}}(1) \oplus {\mathcal{O}}(-1)\).
Theorem: for \(U\) smooth affine \({\mathsf{Sch}_{/k}}\), there is an equivalence of rank \(n\) vector bundles on \(U\) mod equivalence to \(\pi_0 \mathop{\mathrm{Maps}}_{{\mathsf{Spaces}}(k)}(U, {\mathbf{B}}\operatorname{GL}_n)\) where again \({\mathbf{B}}\operatorname{GL}_n \cong {\operatorname{Gr}}_n\).
\begin{align*}\[Algebraic K theory\end{align*} ] : finitely generated projective \(R{\hbox{-}}\)modules mod equivalence with \(\oplus\), then take group completion to get \(K_0(R)\).
To get a space: take \(K(R) \mathrel{\vcenter{:}}=\mathop{\mathrm{Proj}}({\mathsf{R}{\hbox{-}}\mathsf{Mod}})^ {\operatorname{gp} }\) to get a space, set \(K_i R \mathrel{\vcenter{:}}=\pi_i K(R)\)
Prop: the \(K\) theory space here is a motivic space, \(K: {\mathsf{sm}}{\mathsf{Sch}}_{/k}^{\operatorname{op}}\to {\mathsf{Spaces}}\).
Interesting fact: \({\Omega}^\infty {\mathbb{S}}\cong ({\mathsf{FinSet}}, {\textstyle\coprod})^ {\operatorname{gp} }\).
Note from Yuri Sulyma: "B is (widespread but) really bad notation for geometric realization. You should think of B as part of an equivalence \begin{align*} {\mathbf{B}}: \left\{{\text{monoidal categories}}\right\} \to \left\{{\text{pointed connected (2-)categories}}\right\} \end{align*}
up to the \begin{align*}\[Quillen equivalence\end{align*} ]
\begin{align*} {\mathsf{Kan}}\xrightarrow{\sim} {\mathsf{Spaces}} \end{align*}
\begin{align*}\[geometric realization\end{align*} ] takes a (quasi-)category (or \begin{align*}\[simplicial set\end{align*} ]) and inverts all the morphisms.
So \(M^ {\operatorname{gp} } = \Omega{ {\left\lvert {{\mathbf{B}}M} \right\rvert} }\): you take \(M\), \begin{align*}\[deloop\end{align*} ] to turn the objects into morphisms, invert all the morphisms, then take loops to get your objects back."
Theorem ( \begin{align*}\[Morel-Voevodsky\end{align*} ]): \(X\in {\mathsf{sm}}{\mathsf{Sch}_{/k}}\) \begin{align*} K(X) \cong \mathop{\mathrm{Maps}}_{{\mathsf{Spaces}}(k)}(X, {\mathbb{Z}}\times{\operatorname{Gr}}_\infty) .\end{align*}
Uses \begin{align*}\[stratification\end{align*} ] of \({\operatorname{Gr}}\) by affines, thanks \begin{align*}\[Schubert calculus\end{align*} ]!
There is an \({\mathbb{A}}^1\) homotopy equivalence on affines: \(K \simeq{\mathbb{Z}}\times{\operatorname{Gr}}_\infty\). Also, \begin{align*} {\operatorname{Betti}}(k) \simeq{\mathbb{Z}}\times{\operatorname{BU}}= {\Omega}^\infty{\operatorname{KU}} .\end{align*}
Theorem: Can replace \({\operatorname{Gr}}\) with \(\operatorname{Hilb}_\infty({\mathbb{A}}^\infty)\). Very singular!
Definition: \(\operatorname{Hilb}_d({\mathbb{A}}^n)(T)\) are maps \(Z\hookrightarrow{\mathbb{A}}^n\times T\) over \(T\) which are finite \begin{align*}\[flat morphism\|flat\end{align*} ] of degree \(d\) over \(T\).
Morally: \(d{\hbox{-}}\)tuples of points in \({\mathbb{A}}^n\).
Representable! But \(\operatorname{Hilb}_\infty\) is a colimit, thus an \begin{align*}\[Ind scheme\end{align*} ].
This says either the \begin{align*}\[Hilbert scheme\end{align*} ] or \begin{align*}\[K-Theory\end{align*} ] is hard.
In fact the theorem defines a map \({\operatorname{Gr}}_{d-1} \to \operatorname{Hilb}_d({\mathbb{A}}^\infty)\) sending a vector space to the tangent space at 0, and proves this is an \({\mathbb{A}}^1{\hbox{-}}\)homotopy equivalence on affines.
Sends subspace to thick point at zero.
Burt’s proof worked!
\begin{align*}\[Grassmannian\end{align*} ] : parameterizes vector bundles with an embedding into \(\infty\)?
First step in proof: forget embedding into \({\mathbb{A}}^\infty\), send \({ \mathsf{Vect} }_{d-1}\) to finite flat schemes of degree \(d\) \({\mathsf{FFlat}}_d(R)\) over \(\operatorname{Spec}R\), which are stacks.
Send \(V\to R \oplus V\), a \begin{align*}\[square zero extension\end{align*} ], add trivial multiplication.
Inverse: take an algebra \(A\to A/R\) by killing the unit.
Not an equivalence of stacks! Since \(A\not\cong A/R \oplus R\), but the surprising fact is \(A\to A/R \oplus R\) is \({\mathbb{A}}^1\) homotopic to the identity on \({\mathsf{FFlat}}(R)\).
Cook up an explicit homotopy: take the \begin{align*}\[Rees algebra\end{align*} ] \begin{align*} {\operatorname{Rees}}(A) \mathrel{\vcenter{:}}=\left\{{ a_0 + a_1 t + \cdots {~\mathrel{\Big\vert}~}a_0\in R }\right\} \subseteq A[t] .\end{align*}
\({\operatorname{Rees}}(A) / \left\langle{ t-1 }\right\rangle\simeq A\)
\({\operatorname{Rees}}(A) / \left\langle{ t }\right\rangle\simeq R \oplus A/R\).
Tags: #stable_homotopy #seminar_notes
Mike was thinking about computing \begin{align*}\[Topological modular forms\|tmf\end{align*} ] at the prime \(p=3\), since for \(p>3\) it breaks up as a wedge of copies \({\operatorname{BP}}\left\langle{ 2 }\right\rangle\) of \begin{align*}\[Brown-Peterson spectra\end{align*} ]
Roughly twice as hard as computing \begin{align*}\[K-Theory\end{align*} ] with \begin{align*}\[ku\end{align*} ]! (Wilson, Adams, Margalis)
For \(p=2\): an \begin{align*}\[Adams spectral sequence\end{align*} ] (Mahowald, Davis-Mahowald) built out of \begin{align*} H^*( \mathrm{tmf} , {\mathbb{F}}_2) \cong A \otimes_{A(2) } {\mathbb{F}}_2 && \text{where } A(2) = \left\langle{ \operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{Sq}^4 }\right\rangle \end{align*}
For \(p=3\), heuristic: should be like \begin{align*}\[ko\end{align*} ] at \(p=2\) in terms of complexity.
Also thinking about Hopkins-Miller higher real K theories.
First Talbot: huge efforts by Norrah!!!
\begin{align*}\[Formal group\|formal group laws\end{align*} ] over \(R\): a power series \(x +_F y \mathrel{\vcenter{:}}= F(x, y) \in R { \left[ {x, y} \right] }\) such that
A morphism of \begin{align*}\[Formal group\|formal group laws\end{align*} ] : \(f\in R { \left[ {x} \right] }\) with \(f(x+_F y)= f(x) +_G f(y)\).
The functor \(R\to \mathsf{FGL}_{/R}\) is representable, as is the functor sending \(R\) to formal group laws over \(R\) along with an isomorphism \(f\) such that \(f'(0) = 1\).
Theorem (Quillen): \({\operatorname{MU}}_*\) is the ring representing the first functor. See \begin{align*}\[MU\end{align*} ].
Milnor showed \({\operatorname{MU}}_* = {\mathbb{Z}}[x_1, \cdots]\).
How to prove representability: take representing object for power series, check what the conditions translate to.
\({\operatorname{MU}}_* {\operatorname{MU}}\) represents the second factor (i.e. the \({\operatorname{MU}}_*\) homology of \({\operatorname{MU}}\), given by \(\pi_*({\operatorname{MU}}\wedge{\operatorname{MU}}))\).
Example: if \(n\in {\mathbb{N}}\), then \begin{align*} [n]_F (x) = \overset{F}{\sum_{k\leq n}} x = nx + \cdots \end{align*} is an endomorphism of \(F\).
If \(\char R = p\), then \([p]_F (x) = f(x^{p^n})\), if \(f'(0) \in R^{\times}\) then the \(\operatorname{ht}F=n\) and \(f(x) = v_n x + \cdots\).
For \(R\) a field, it’s a theorem that the \begin{align*}\[height of a formal group law\end{align*} ] is a complete invariant for algebraically closed fields.
Picture of the \begin{align*}\[moduli of formal group laws\end{align*} ] :
\begin{align*}\[attachments/image_2021-06-06-15-32-17.png\end{align*} ]
How to glue: sheaf condition on opens? Extensions on closed sets? But how do you talk about gluing an open to a closed set?
Explained by \begin{align*}\[deformation theory\end{align*} ] : can push not only in direction in the space, but also into the tangent space directions.
\begin{align*}\[deformation\end{align*} ] : a ring map \(A\to k\) with a nilpotent kernel:
\begin{align*}\[attachments/image_2021-06-06-15-35-33.png\end{align*} ]
\begin{align*}\[attachments/image_2021-06-06-15-37-51.png\end{align*} ]
Here \(\widehat{G}\) sends a ring \(R\) to the set of nilpotent elements of \(R\), and \(F\) gives that a group structure – the algebraic geometry gadget corresponding to the formal group law \(F\).
Examples:
Theorem (Lubin-Tate): there is a universal deformation for \((k, F)\) given by \begin{align*} {\mathbb{W}}(k) { \left[ {{ {u}_1, {u}_2, \cdots, {u}_{n-1}}} \right] } \mathrel{\vcenter{:}}= E(k, F)_0 .\end{align*}
See \begin{align*}\[Lubin-Tate theory\end{align*} ] and \begin{align*}\[Witt Vectors\|Witt vector\end{align*} ].
For \(k = {\mathbb{F}}_p\), we have \({\mathbb{W}}(k) = { {\mathbb{Z}}_p }\), and there is an action of \(\mathop{\mathrm{Aut}}(F)\) and \({ \mathsf{Gal}} (k)\).
Theorem ( \begin{align*}\[Goerss-Hopkins-Miller\end{align*} ]): there is a canonical functor \((k, F) \to E(k, F)\) such that
This lifts the AG problem to a problem in commutative \begin{align*}\[ring spectra\end{align*} ].
Theorem (Devinats-Hopkin?): the map \begin{align*} L_{K(n)} S^0 \xrightarrow{\simeq} E_n^{h{\mathbb{G}}_n} \end{align*} is an equivalence where \begin{align*} E_n &\mathrel{\vcenter{:}}= E({\mathbb{F}}_{p^n}, F_{\mathrm{Honda}}) \\ {\mathbb{G}}_n &\mathrel{\vcenter{:}}={ \mathsf{Gal}} ({\mathbb{F}}_{p^n}) \rtimes S_n \\ S_n &\mathrel{\vcenter{:}}=\mathop{\mathrm{Aut}}(F_{\mathrm{Honda}}) .\end{align*}
Define \begin{align*}\[Hopkins-Miller higher real K theories\end{align*} ] : for \(G \subseteq S_n\) finite, \({\mathsf{E} {\operatorname{O}}}_n(G) = E_n^{hG}\).
Example: for \(n=1, p=2\) we have \({\mathsf{E} {\operatorname{O}}}_1(C_2) = {\operatorname{KO}}{ {}^{ \widehat{2} } }\), which is completely understood via the \begin{align*}\[homotopy fixed point spectral sequence\end{align*} ].
Example: for \(n=2\), we know \begin{align*} {\mathsf{E} {\operatorname{O}}}_2(G) = L_{K(2)} \mathrm{tmf} \end{align*} or a summand thereof.
Use Adams indexing, look at homotopy fixed point spectral sequence \begin{align*} H^s(G; \pi_t E_n)\Rightarrow\pi_{t-s} {\mathsf{E} {\operatorname{O}}}_n(G) .\end{align*}
We know a lot about the bottom row: \(H^0(G; \pi_* E_n) = (\pi_* E_n)^G\), using that \begin{align*}\[group cohomology\end{align*} ] is the derived functor of \(G{\hbox{-}}\)invariants.
If \({\sharp}G\) is prime to \(p\) then \(H^{> 0}(G, \pi* E_n) = 0\).
We don’t know much else about anything in this spectral sequence!
Consider \begin{align*} {\mathcal{O}}_n \mathrel{\vcenter{:}}={\mathbb{W}}({\mathbb{F}_{p^n}}) \left\langle{ s }\right\rangle / \left\langle{ sa = a^{ \varphi} S }\right\rangle \end{align*} where \(\varphi\) is the Frobenius on \({\mathbb{F}_{p^n}}\).
This is the \begin{align*}\[Dieudonne module\end{align*} ] for \(F_\mathrm{\operatorname{Honda}}\)?
Power series rings have a maximal ideal \({\mathfrak{m}}\), so consider \begin{align*} \pi_{-2} E_n / \left\langle{ p, {\mathfrak{m}}^2}\right\rangle .\end{align*}
There is an \({\mathcal{O}}_n^{\times}\) \begin{align*}\[equivariant\end{align*} ] map from \({\mathcal{O}}_n/p\) to this, and the question is how to lift:
\begin{align*}\[attachments/image_2021-06-06-15-58-57.png\end{align*} ]
Devinats-Hopkins: up to \begin{align*}\[associated graded\end{align*} ], \begin{align*} \pi_* E_n \cong \operatorname{Sym}({\mathcal{O}}_n) \left[ { \scriptstyle { {\Delta}^{-1}} } \right]{ {}^{ \widehat{I} } } .\end{align*}
Use that the symmetric algebra is a free thing.
Warning: can’t make this \(S_n\) \begin{align*}\[equivariant\end{align*} ], so can’t compute the whole thing using this approach.
“Up to associated graded” means there’s a spectral sequence relating.
Mike thought about this a lot in grad school! But wound up doing his these on the first topic about \(H^*\) of \begin{align*}\[tmf\end{align*} ].
2007 Talbot: Mike Hopkins was the faculty mentor.
Theorem ( \begin{align*}\[Hill-Hopkins-Ravenel\end{align*} ]): there is a \(G{\hbox{-}}\)equivariant lift \({\mathcal{O}}_n \to \pi_{-2} E_n\) for any finite \(G\).
Real K theories see a lot of \(\pi_* {\mathbb{S}}\).
Theorem (Ravenel): for \(p\geq 5\), some element \(\beta_{p^i / p^i}\) does not survive the \begin{align*}\[ANSS\end{align*} ].
Exact same argument for \begin{align*}\[Kervaire invariant 1\end{align*} ] : \(p=2\) version of this argument?
\({ \underset{(\infty, {2})}{ \mathsf{Cat}} }\): discrete set of objects, enriched in categories, \(2{\hbox{-}}\)morphisms are strictly associative?
\({ \underset{(\infty, {n})}{ \mathsf{Cat}} }\): all \(n+k{\hbox{-}}\)morphisms are invertible.
There is an embedding \(n{\hbox{-}}\mathsf{Cat}\hookrightarrow{ \underset{(\infty, {n})}{ \mathsf{Cat}} }\)with a specific model structure.
Model structure on \({\mathsf{sSet}}\): fibrant objects are Kan complexes, there is a Quillen equivalence between \({\mathsf{Top}}\) and \({\mathsf{sSet}}\)
\({ \underset{(\infty, {1})}{ \mathsf{Cat}} }\): enriched in \({ \underset{\infty}{ \mathsf{Grpd}} }\)
\begin{align*}\[attachments/image_2021-06-05-12-16-29.png\end{align*} ]
\begin{align*}\[Joyal model structure\end{align*} ] : \({\mathsf{sSet}}\) with quasicategories as fibrant objects.
\begin{align*}\[Quillen equivalence\end{align*} ] to the \begin{align*}\[Kan model structure\end{align*} ] by taking the \begin{align*}\[homotopy coherent nerve\end{align*} ]?
Exercise: \begin{align*}\[nerve\end{align*} ] is a \begin{align*}\[Kan complex\end{align*} ] iff \(\mathsf{C}\) is a \begin{align*}\[groupoid\end{align*} ].
Complete \begin{align*}\[Segal space\end{align*} ] : a \begin{align*}\[simplicial space\end{align*} ] \(W \Delta^{\operatorname{op}}\to {\mathsf{sSet}}_{{\mathsf{Kan}}}\) with some conditions.
\begin{align*}\[simplicial set\end{align*} ] \({\mathsf{sSet}}^{\Delta^{\operatorname{op}}}\) has a model structure where complete Segal spaces are the fibrant objects
Exercise: if \(W\) is a complete Segal space then \(i_1^* W\) is a quasicategory where \(i_1: \Delta \to \Delta^{\times 2}\) sends \([n]\) to \(([n], [0])\).
Limits: isomorphisms on hom sets, or terminal objects in the \begin{align*}\[cone category\end{align*} ] :
\begin{align*}\[attachments/image_2021-06-05-12-24-46.png\end{align*} ]
Terminal objects: isomorphisms from \begin{align*}\[slice category\end{align*} ] to \(\mathsf{C}\):
\begin{align*}\[attachments/image_2021-06-05-12-25-36.png\end{align*} ]
Enriched definition of limits:
\begin{align*}\[attachments/image_2021-06-05-12-27-23.png\end{align*} ]
Terminal objects: \(X_{/ {x}} \xrightarrow{\sim} X\) is a \begin{align*}\[weak equivalence\end{align*} ] in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\).
All definitions of limits in \({ \underset{(\infty, {1})}{ \mathsf{Cat}} }\) recover the usual notions via the nerve.
Exercise: Show that the limit of \(F: I\to \mathsf{C}\) is the limit of its \begin{align*}\[nerve\end{align*} ]?
Theorem: limit of \(F\) in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\) is a \begin{align*}\[homotopy limit\end{align*} ] of its \begin{align*}\[adjoint\end{align*} ] in \({\mathsf{sSet}}{\hbox{-}}\mathsf{Cat}_{{\mathsf{Kan}}}\), and the limit of its adjoint in \({\mathsf{sSet}}^{\Delta^{\operatorname{op}}}_{ \mathsf{CSS} }\).
Upshot: many different models, can move between different models.
Things to look up from written notes:
Next: models of \({ \underset{\infty}{ \mathsf{Cat}} }{\infty, 2}\)
\begin{align*}\[base change along a functor\end{align*} ]??
\begin{align*}\[2-category\end{align*} ] : categories \begin{align*}\[enriched\end{align*} ] in categories
Recall: \({ \mathsf{CSS} } = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{sSet}})\).
Idea: keep track of 2-morphisms, i.e. two-cells, can keep all of their possible compositions
\begin{align*}\[attachments/image_2021-06-05-13-09-28.png\end{align*} ]
\begin{align*}\[attachments/image_2021-06-05-13-09-54.png\end{align*} ]
\begin{align*}\[attachments/image_2021-06-05-13-10-05.png\end{align*} ]
Problem: for many \begin{align*}\[QFT\|QFTs\end{align*} ], we don’t know how to write down the quantum \begin{align*}\[observables\end{align*} ] \({\mathcal{F}}(U)\) for an open \(U \subseteq X\) (e.g. for \(X\) spacetime).
Three approaches:
Factorizable cosheaves (topological/differential geometric) Quantum observables in the field theory.
Vertex algebras (algebra and analysis) Infinite dimensional vector spaces, symmetries of \(2d\) conformal field theories
Chiral or \begin{align*}\[factorization algebra\|factorization algebras\end{align*} ] (algebraic geometry)
\begin{align*}\[quasicoherent sheaf\|quasicoherent sheaves\end{align*} ] (so \begin{align*}\[D modules\end{align*} ]) with \begin{align*}\[Lie algebra\|Lie (co)algebra\end{align*} ] structures. Collisions between local operators
\begin{align*}\[vertex algebra\end{align*} ], meromorphic multiplication \(V^{\otimes 2} \to V((z))\).
\begin{align*}\[vertex operator\end{align*} ] : \(Y({-}, z): V\to \mathop{\mathrm{End}}V { \left[ {z, z^{-1}} \right] }\) where \(A\mapsto \sum A_{(n)} z^?\) where \(A_{(n)}:V\to V\) should be thought of as ways of multiplying.
Any commutative algebra with a \begin{align*}\[derivation\end{align*} ] \(T\) yields a vertex algebra \(Y(A, z) = e^{zT} A = \sum _{T^k A \over k!} z^k\).
The \begin{align*}\[monster group\end{align*} ] is the largest sporadic simple group, constructed as the automorphisms of a \begin{align*}\[vertex algebra\end{align*} ] constructed from the \begin{align*}\[Leech lattice\end{align*} ].
We knew the dimensions of representations before the construction (e.g character tables), conjectured to be related to \begin{align*}\[modular functions\end{align*} ], Borcherds Fields in 98 for proving this!
Important fact: certain categories of representations of affine Lie algebras/quantum groups form \begin{align*}\[modular tensor categories\end{align*} ] : Kazhdan-Lusztig 93!
Nice invertible objects? Levels are closed under tensor?
Beilinson-Drinfeld, 90s: factorization/chiral algebras
Discrete example: particles on a surface labeled with integers, where colliding causes addition of labels.
Ex: the \begin{align*}\[Hilbert scheme\end{align*} ] of points. Lengths of subschemes equals dimension of quotient of \(\operatorname{Spec}\) as a vector space over \({\mathbb{C}}\). Consider \(\operatorname{Spec}{\mathbb{C}}[x, y] / \left\langle{ x, y(y- \lambda) }\right\rangle\). \(\lambda=0\) remembers that the collision happened along the \(y\) axis.
\(\operatorname{Hilb}_X\) is smooth when \(\dim X = 1,2\).
Most important example of a factorization space: \begin{align*}\[Beilinson-Drinfeld Grassmannian\end{align*} ].
Upshot: combine all 3 approaches to tackle problems!
\begin{align*}\[vertex algebra\end{align*} ] : a factorization algebra over curves (with more symmetry)
A vertex algebra is quasi-conformal if it has a nice action of \(\mathop{\mathrm{Aut}}\operatorname{Spf}{\mathbb{C}} { \left[ {t} \right] }\), automorphisms of a formal disk? See \begin{align*}\[formal spectrum\end{align*} ].
(Corresponds to Virasoro symmetry of the CFT).
Note: aut of formal disk is more like in ind object in \begin{align*}\[group scheme\|Group schemes\end{align*} ]?
Not an \begin{align*}\[algebraic group\end{align*} ], carries some limits/colimits?
Direct bridge from \begin{align*}\[factorizable cosheaves\end{align*} ] to factorizable algebras doesn’t quite exist yet!
Reference: eAKTs
Tags: #seminar_notes #k_theory
Refs:
\begin{align*}\[Brauer group\end{align*}
]
\begin{align*}\[Azumaya algebra\end{align*}
]
This is some \(H^2\) perhaps? Like \(\mathop{\mathrm{Br}}(X) = H^2(X; {\mathbb{G}}_m)\)? Need to figure out what kind of cohomology this is though.
See \begin{align*}\[Brauer-Manin pairing\end{align*} ], \begin{align*}\[Tate pairing\end{align*} ]
degree of cycles on \begin{align*}\[Chow ring\|Chow\end{align*} ]
See \begin{align*}\[central fiber\end{align*} ], \begin{align*}\[formal scheme\end{align*} ]
\(\lim^1\), see \begin{align*}\[lim1\end{align*} ]
See \begin{align*}\[GAGA\end{align*} ]
Morita theory: for \(R\in \mathsf{Ring}, A,B\in {\mathsf{Alg}}_{R}\), \(A\sim B\) are \begin{align*}\[Morita equivalent\end{align*} ] iff \({\mathsf{A}{\hbox{-}}\mathsf{Mod}} \equiv \mathsf{B}{\hbox{-}}\mathsf{Mod}\), and \(A\) is \begin{align*}\[Azumaya algebra\|Azumaya\end{align*} ] if it’s \begin{align*}\[invertible object of a category\end{align*} ] in the following sense: there is an \(A'\) such that \(A\otimes A' \sim R\)
What is \begin{align*}\[presentable infinity category\end{align*} ]?
Part of an equivalence: take a \begin{align*}\[compact generator\end{align*} ], take its endomorphism algebra, take category of modules over that algebra?
See \begin{align*}\[etale descent\end{align*} ] and \begin{align*}\[Zariski descent\end{align*} ].
\begin{align*}\[invertible object of a category\end{align*} ] implies \begin{align*}\[dualizable object of a category\end{align*} ] but not conversely.
Smooth and proper: dualizable?
See \begin{align*}\[perfect complexes\|perfect complex\end{align*} ]
Reference: ???, GROOT
Big idea: free implies flat for algebras, is this true in the equivariant settings?
Abelian groups \(\approx\) \begin{align*}\[Mackey functor\end{align*} ].
\({\mathbb{Z}}\approx\) Burnside Mackey functors
Commutative rings \(\approx\) \begin{align*}\[Green functor\end{align*} ] \((E_\infty\) algebras), Incomplete functors, \begin{align*}\[Tambara functor\end{align*} ]
Free algebra \({\mathbb{Z}}[G]\) comparable to free incomplete Tambara functor
Similarities come from being algebras over \begin{align*}\[Operads\end{align*} ].
\begin{align*}\[Hill-Hopkins-Ravenel\end{align*} ] involves spectral sequences of Mackey functors
All rational Mackey functors are free
\(A^{\mathcal{O}}[x_{G/H}]\) is almost never flat.
A Mackey functor is an additive functor \(M: A^g\to {\mathsf{Ab}}\), where \(A^G\) is the \begin{align*}\[Burnside category\end{align*} ] : finite \(G{\hbox{-}}\)sets, where morphisms \(A^G(X, Y)\) is the group completion wrt \(\coprod\) of finite \(G{\hbox{-}}\)sets, so \begin{align*}\[spans\end{align*} ].
To define a Mackey functor \(F\), it suffices to give abelian groups \(F(G/H)\) for \(H\leq G\), restrictions \(\operatorname{res}^H_K\), and \begin{align*}\[transfer map\end{align*} ] \({\mathrm{tr}}_K^H\) in the target.
\begin{align*}\[Burnside Mackey functor\end{align*} ] : \(\underline{A}\).
Theorem (Lewis): the category of Mackey functors is abelian, and has a \begin{align*}\[symmetric monoidal category\|symmetric monoidal\end{align*} ] product \(\boxtimes\) with unit \(\underline{A}\).
A \begin{align*}\[Green functor\end{align*} ] is a monoid for \(\boxtimes\), which is an \begin{align*}\[E_n ring spectrum\|E_infty algebra\end{align*} ] in Mackey functors.
An incomplete Tambara functor is an \(N_\infty\) algebra in Mackey functors
A \begin{align*}\[Tambara functor\end{align*} ] is a \begin{align*}\[Green functor\end{align*} ] with that data of a \begin{align*}\[norm\end{align*} ] map \(\nm_K^H\), a multiplicative morphism.
Indexing systems: valid suborderings on the poset lattice of subgroups
Theorem (Barnes-Roitzheim-?) For \(C_{pq}\), there are roughly a \begin{align*}\[Catalan's number\end{align*} ] of valid indexing systems.
Tags: #idle_thoughts
Really cool idea I like from that talk: what is the probability density of objects in a category?
In a precise sense, what proportion of objects are projective, flat, free, dualizable, indecomposable, simple, etc?
I think I really want analytic structure on a category! I’m reminded of results like \begin{align*}\[Morse functions\end{align*} ] being generic in spaces of functions, or perturbing \begin{align*}\[Hamiltonian\|Hamiltonians\end{align*} ] in \begin{align*}\[Floer Theory\end{align*} ]. We can cook up topologies to make these kinds of statements precise in the classical setting….how can we do it here?
Look up \begin{align*}\[simple normal crossings\end{align*} ] divisor.
The \begin{align*}\[Gelfand representation\end{align*} ] is really cool. Look into how this duality shows up for schemes!
What is the \begin{align*}\[length of a module\end{align*} ]?
Tags: #idle_thoughts #category_theory #infinity_cats
Terrible attempt at a way around ZFC: axiomatically define a category the way one axiomatizes Euclidean geometry:
This makes the definition infinitely recursive, which might be a problem. One could truncate this by asking the 1st iteration of taking “the hom category” to result in a discrete category: some objects but no morphisms between distinct objects. This is definitely taken care of by \begin{align*}\[infinity categories\end{align*} ].
Define the category of sets by specifying a single point as an initial object, then freely taking powersets and unions. I think you at least get something whose nerve is the same as the nerve of the category of finite sets. I think one can also realize these operations at the categorical level: powersets are like exponentials \(2^X\), you can get disjoint unions from limits, and maybe usual unions/intersections from pushouts/pullbacks?
Tags: #seminar_notes #geometric_topology #floer Refs: \begin{align*}\[Heegard-Floer homology\end{align*} ]
Reference: Kristen Hendricks, Surgery formulas for involutive Heegaard Floer homology. Stanford Topology Seminar.
Want to study homology \begin{align*}\[cobordism groups\end{align*} ] of \begin{align*}\[3-manifolds\end{align*} ] \(\Theta_{\mathbb{Z}}^3\).
See \begin{align*}\[Seifert fibered spaces\end{align*} ].
People like \begin{align*}\[HF\end{align*} ] because there are a lot of computational tools! In particular, a \begin{align*}\[surgery formula\end{align*} ].
Check out \begin{align*}\[ideal sheaves\end{align*} ], \begin{align*}\[Birdgeland stability conditions\end{align*} ].
“Rotate” an exact sequence to get an \begin{align*}\[exact triangle\end{align*} ] :
What is an \begin{align*}\[etale algebra\end{align*} ]?
See \begin{align*}\[Serre's uniformity conjecture\end{align*} ].
Source: Frobenius exact symmetric tensor categories - Pavel Etingof. IAS Geometric/modular representation theory seminar. https://www.youtube.com/watch?v=7L06K7SL5qw
Tags: #seminar_notes #representation_theory #category_theory #monoidal Refs: \begin{align*}\[tensor category\end{align*} ]
Looking at \begin{align*}\[modular representations\end{align*} ] of finite groups.
See \begin{align*}\[tensor ideal\end{align*} ], \begin{align*}\[Krull-Schmidt theorem\end{align*} ] : decomposition into indecomposables is essentially unique.
Can take \begin{align*}\[split Grothendieck ring\end{align*} ].
\begin{align*}\[Symmetric tensor category\end{align*} ] \(\mathsf{C}\):
Example: \({\mathsf{Rep}}_k(G)\) the category of finite-dimensional representations of \(G\). Take \(G=1\) to recover \({ \mathsf{Vect} }_{/k}\).
Such a category is \begin{align*}\[tannakian\end{align*} ] if there exists a \begin{align*}\[fiber functor\end{align*} ] : a symmetric tensor functor \(F: \mathsf{C} \to { \mathsf{Vect} }_{/k}\).
For an ( \begin{align*}\[additive category\end{align*} ] rigid symmetric monoidal category \(\mathsf{C}_{/k}\) with \(\mathop{\mathrm{End}}_{\mathsf{C}}(\one) = k\), for any \(f\in \mathsf{C}(X, X)\) we can define its \begin{align*}\[trace (monoidal categories)\|trace\end{align*} ] \(\operatorname{Tr}(f) \in \mathop{\mathrm{End}}_{\mathsf{C}}(\one)\):
What is an \begin{align*}\[isocrystal\end{align*} ]?
What does it mean to be \begin{align*}\[crystalline\end{align*} ]?
What is a \begin{align*}\[symplectic basis\end{align*} ]?
What is the \begin{align*}\[Mordell-Weil sieve\end{align*} ]?
How can one pass from p-adic solutions to rational solutions?
What is the \begin{align*}\[conductor of an elliptic curve\end{align*} ]?
What is \begin{align*}\[Serre's open image theorem\end{align*} ].
What is \begin{align*}\[inertia\end{align*} ]?
What is the \begin{align*}\[cyclotomic character\end{align*} ]?
See \begin{align*}\[Mazur Program B\end{align*} ]
See \begin{align*}\[Goursat's lemma in group theory\end{align*} ].
We haven’t been able to classify the rational points on \begin{align*}\[modular curves\end{align*} ]!
Reference: Kirsten Wickelgren, Colloquium Presentation: zeta functions and a quadratic enrichment. Rational Points and Galois Representations workshop
Tags: #seminar_notes #homotopy_theory #algebraic_geometry #stable_homotopy #motivic Refs: \begin{align*}\[motivic homotopy\end{align*} ]
See \begin{align*}\[Atiyah duality\end{align*} ] : define the dual of \(M\) as \(M^{-{\mathbf{T}}M}\), the \begin{align*}\[Thom space\|Thom space\end{align*} ] of (minus) the tangent bundle.
Define the \begin{align*}\[trace (monoidal categories)\|trace\end{align*} ] :
Then \(\operatorname{Tr}(\phi) \in \mathop{\mathrm{End}}_{\mathsf{C}}(\one, \one)\) is an endomorphism of the unit.
Example: \begin{align*}\[Lefschetz fixed point theorem\end{align*} ], \begin{align*} \operatorname{Tr}(\phi) = \sum_{x\in M, \phi(x) = x} \operatorname{Ind}_x \phi \in \mathop{\mathrm{End}}_{{\mathsf{ho}}{\mathsf{Sp}}}(\one) \xrightarrow{\deg \,\, \sim} {\mathbb{Z}} ,\end{align*} where we take the degree of a map between spheres.
\({\mathbb{S}}= \one \in {\mathsf{ho}}{\mathsf{Sp}}\).
Use that \(H^*({-}, {\mathbb{Q}})\) preserves tensor products, and apply the Kunneth formula: \begin{align*} H^*(\operatorname{Tr}(\phi)) &= \operatorname{Tr}(H^*(\phi)) \\ \implies \sum (-1)^i \operatorname{Tr}( H^i(\phi); H^i(M) {\circlearrowleft}) &= \sum_{x\in M, \phi(x) = x} \operatorname{Ind}_x \varphi .\end{align*}
Rationality of \(\zeta\):
Use hocolims to glue spaces, but may not work in schemes.
Example: take \(X \mathrel{\vcenter{:}}={\mathbb{P}}^n/{\mathbb{P}}^{n-1}\), then we’d want \(X({\mathbb{C}}) \cong S^{2n}\) and \(X({\mathbb{R}}) \cong S^n\)
Problem: this quotient isn’t a \begin{align*}\[scheme\end{align*} ]. Can freely add these limits.
We want \({\mathbb{P}}^i / {\mathbb{P}}^{i-1}\) to be the building blocks or cells
Morel and Voevodsky, \({\mathbb{A}}^1\) \begin{align*}\[stable homotopy category\end{align*} ] over \(k\), denoted \({\mathsf{SH}}(k)\).
Take an analog of degree, the Morel degree: \begin{align*} \deg: [{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1} ] \xrightarrow{} {\operatorname{GW}}(k) .\end{align*}
\begin{align*}\[Grothendieck-Witt\end{align*} ] group: formal differences of isomorphism classes of nondegenerate symmetric \begin{align*}\[bilinear forms\end{align*} ].
Special form: the hyperbolic form \begin{align*} \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle = \left\langle{a}\right\rangle + \left\langle{-a}\right\rangle = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} .\end{align*}
See \begin{align*}\[rank\end{align*} ], \begin{align*}\[signature\end{align*} ], \begin{align*}\[discriminant\end{align*} ] of \begin{align*}\[quadratic form\|quadratic forms\end{align*} ].
Trace here will take values in \({\operatorname{GW}}(k)\).
\begin{align*}\[Lefschetz fixed point theorem\end{align*} ] due to Hoyois:
Notation: \(dZ^{{\mathbb{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log \zeta^{{\mathbb{A}}^1}(t) = \sum_{m\geq 1} \operatorname{Tr}(\phi^m)t^{m-1}\).
Prop: \begin{align*} \operatorname{rank}dZ^{{\mathbb{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log .\end{align*}
See Kapranov \begin{align*}\[motivic zeta function\end{align*} ] :
See \begin{align*}\[Euler class\end{align*} ], \begin{align*}\[Hopf map\end{align*} ]
Major point: this is genuinely something new, isn’t just recovered by taking the compactly supported euler characteristic:
Tags: #THH #factorization_homology #seminar_notes Refs: \begin{align*}\[nonabelian Poincare duality\end{align*} ], \begin{align*}\[factorization homology\end{align*} ]
Reference: Foling Zou, Nonabelian Poincare duality theorem and equivariant factorization homology of Thom spectra. MIT Topology Seminar.
Goal: want to formulate \begin{align*}\[monads\end{align*} ] and \begin{align*}\[operads\end{align*} ] categorically.
See \begin{align*}\[lambda sequences\end{align*} ], something like a functor \({\mathsf{FinSet}}^{\operatorname{op}}\to \mathsf{C}\)?
See \begin{align*}\[Day convolution\end{align*} ] as an example of a monoidal product.
See \begin{align*}\[monadic bar construction\end{align*} ] and monoidal \begin{align*}\[bar construction\end{align*} ].
Examples of \begin{align*}\[factorization homology\end{align*} ] : \begin{align*} \int_{S^1}A &&\simeq{\operatorname{THH}}(A) \\ \int_{T^n}A &&\simeq{\operatorname{THH}}^n(A) && \text{iterated THH} .\end{align*}
For \(\sigma\) the \begin{align*}\[sign representation\end{align*} ], \(\int_{S^\sigma} A \simeq\operatorname{THR}(A)\) for \(E_\sigma{\hbox{-}}C_2\) spectra.
See Horev, Hessolholt-Madsen.
Axiomatic approach to factorization homology: take a left \begin{align*}\[Kan extension\end{align*} ] of the following:
Can compute \begin{align*}\[Kan extension\end{align*} ] via the \begin{align*}\[bar construction\end{align*} ].
Theorem: \begin{align*}\[equivariant\end{align*} ] \begin{align*}\[nonabelian Poincare duality\end{align*} ] :
What is the \begin{align*}\[virtual dimension\end{align*} ] of a bundle?
\({\operatorname{Pic}}\): subcategory of invertible objects, \begin{align*}\[Picard group\|PIcard groupoid\end{align*} ]
\begin{align*}\[Thom spectrum\end{align*} ] functor:
where \(R{\hbox{-}}\)line is the \(\infty{\hbox{-}}\)category of line bundles up to equivalence?
Preserves \(G{\hbox{-}}\)colimits, so formally the Thom spectrum functor commutes with factorization homology.
In proof of theorem, use \begin{align*}\[nonabelian Poincare duality\end{align*} ] to reduce a complicated gadget to a mapping space.
For \(\operatorname{THR}\) on the algebra side, see Teena Gerhardt’s work? Haynes Miller suggests looking at the \begin{align*}\[de Rham-Witt complex\end{align*} ]?
Not every \begin{align*}\[simplicial complex\end{align*} ] is a \begin{align*}\[PL\end{align*} ] manifold:
\begin{align*}\[Arpon Raksit - Hochschild homology and the derived de Rham complex revisited\end{align*} ]
\begin{align*}\[Andrew Blumberg, Floer homotopy theory and Morava K-theory\end{align*} ]
\begin{align*}\[Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups\end{align*} ]
^22ba3a
Reference: Padmavathi Srinivasan, UGA NT Seminar.
If \(r\) is the rank and \(g\) is the genus for \(X\) a nice curve over \({\mathbb{Q}}\) with good reduction at \(p\),
See p-adic \begin{align*}\[height pairing\end{align*} ].
Can compute heights as intersection numbers on regular models.
See \begin{align*}\[Abel-Jacobi map\end{align*} ]
Try to realize rational points as the zero locus of p-adic analytic functions.
Recent work: quadratic Chabauty used to find all rational points on the infamously cursed \begin{align*}\[modular curve\end{align*} ] \(X_S(13)\).
Looking for explanations through \begin{align*}\[Arakelov theory\end{align*} ] instead of \begin{align*}\[p-adic Hodge theory\end{align*} ].
See \begin{align*}\[Mordell-Weil rank\end{align*} ]
See \begin{align*}\[Néron-Severi\end{align*} ] of the \begin{align*}\[Jacobian\end{align*} ].
Get height functions where local heights \(h_p\) can be computed by iterated \begin{align*}\[Coleman integrals\end{align*} ].
There is a canonical height machine for abelian varieties.
Symmetric line bundles: \(\mathcal{L}\cong [1]^* \mathcal{L}\).
Zhang defines a \begin{align*}\[metric on a line bundle\end{align*} ], which gives a way to measure the size of elements in each fiber.
When you have an integral model: take closures!
See \begin{align*}\[valuations\end{align*} ], used to define admissible metrics.
Admissible metrics on \({\mathcal{O}}_X\) factor through the \begin{align*}\[reduction graph\end{align*} ]
Adelic metric: a collection of metrics for almost every \begin{align*}\[place\end{align*} ].
Rigidified bundles: remember a point in the fiber.
For each place, there is a canonical metric which makes certain isomorphisms into isometries in a \begin{align*}\[Banach space\end{align*} ]
Curvature form \(\mathop{\mathrm{Curv}}(\mathcal{L}_v)\) is sent to the first \begin{align*}\[Chern class\end{align*} ] \(c_1(\mathcal{L}_v )\) under cup product in de Rham cohomology.
To get a canonical metric for all line bundles, it suffices to canonically metrize the \begin{align*}\[Poincare bundle\end{align*} ]. Every line bundle on \(A\) is a pullback of it.
Any two metrics with the same curvature differ by \(\int \omega\) for \(\omega\).
Reference: The reciprocity law for the twisted second moment of Dirichlet L-functions https://arxiv.org/pdf/0708.2928.pdf
What is a \begin{align*}\[Dirichlet character\end{align*} ]?
What is a \begin{align*}\[Gauss sum\end{align*} ]?
What is the completion of an \begin{align*}\[L function\end{align*} ]? Guessing this has to do with continuation.
What is Dirichlet’s trick?
How can you break a sum up into \begin{align*}\[arithmetic progressions\end{align*} ]?
Reference: The \(K'\)-theory of \begin{align*}\[monoid\end{align*} ] sets https://arxiv.org/pdf/1909.00297.pdf
\begin{align*}\[K-Theory\end{align*} ]
The category \({\mathsf{FinSet}}_{{\scriptstyle { * } }}\) (see \begin{align*}\[Finset\end{align*} ] ) of finite pointed sets is quasi-exact, and \begin{align*}\[Barratt-Priddy-Quillen\end{align*} ] implies that \(K({\mathsf{FinSet}}_{\scriptstyle { * } }) \simeq{\mathbb{S}}\).
\begin{align*}\[Group completion\end{align*} ] : comes from \({\Omega}^\infty {\Sigma}^\infty {\mathbf{B}}G_+\).
Big theorem: \begin{align*}\[Devissage\end{align*} ]. But I have no clue what this means. Seems to say when \({\mathsf{K}}(A) \cong {\mathsf{K}}(B)\)?
Apparently easy theorem: \({\mathsf{K}}'({\mathbb{N}}) \simeq{\mathbb{S}}\).
The \begin{align*}\[Picard group\end{align*} ] of \({\mathbb{P}}^1\) shows up:
Tags: #seminar_notes #algebraic_geometry
Reference: Stefan Schreieder, Refined unramified cohomology. Harvard/MIT AG Seminar talk.
See the \begin{align*}\[Chow ring\end{align*} ] and \begin{align*}\[cycle class map\end{align*} ]. Understanding the image amounts to the \begin{align*}\[Hodge conjecture\end{align*} ] and understanding torsion in the image \(Z^i(X)\)?
\begin{align*}\[Gysin sequence\end{align*} ] yields a residue map \({\partial}_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)\).
Interesting parts of the Coniveau spectral sequence: something coming from unramified cohomology, and something coming from algebraic cycles mod algebraic equivalence.
Failure of integral \begin{align*}\[Hodge conjecture\end{align*} ] :
Uses \begin{align*}\[Bloch-Kato conjecture\end{align*} ]
Allows detecting classes in \(Z^2(X)\) using \begin{align*}\[K-Theory\end{align*} ] methods.
See \begin{align*}\[Borel-Moore cohomology\end{align*} ] – for \(X\) a smooth \begin{align*}\[algebraic scheme\end{align*} ], essentially singular homology with a degree shift?
See \begin{align*}\[pro-objects\end{align*} ] and \begin{align*}\[ind-objects\end{align*} ] in an arbitrary category.
Filter by codimension, then obstructions to extending over higher codimension things is measured by cohomology of the \begin{align*}\[Function field\end{align*} ] :
Here \({{\partial}}\) is a residue map.
See \begin{align*}\[separated\end{align*} ] schemes of \begin{align*}\[finite type\end{align*} ].
Main theorem, works not just for smooth schemes, but in greater generality:
Torsion in the Griffiths group is generally not finitely generated.
See \begin{align*}\[canonical class\end{align*} ] \(K_S\) for a surface, \begin{align*}\[Abel-Jacobi invariants\end{align*} ]?
No Poincaré duality for Chow groups, at least not at the level of cycles. Need to pass to cohomology.
Theorem: there exists a \begin{align*}\[regular\end{align*} ] \begin{align*}\[flat morphism\end{align*} ] \begin{align*}\[proper\end{align*} ] \(S\to \operatorname{Spec}{\mathbb{C}}{\left[\left[ t \right]\right] }\) such that \(S_\eta\) is an Enriques surface, \(S_0\) is a union of \begin{align*}\[ruled surfaces\end{align*} ], and \(\mathop{\mathrm{Br}}(S) \twoheadrightarrow\mathop{\mathrm{Br}}(S_\eta)\).
See \begin{align*}\[Zariski locally\end{align*} ] and \begin{align*}\[étale locally\end{align*} ].
\begin{align*}\[unramified cohomology\end{align*} ] is linked to \begin{align*}\[Milnor K theory\end{align*} ].
Reference: https://www.youtube.com/watch?v=XTOwj1LvntM
Clausen: a baby topic in \begin{align*}\[geometric representation theory\end{align*} ] is \begin{align*}\[Springer correspondence\end{align*} ].
Tags: #representation_theory
See \begin{align*}\[Hopf algebra\end{align*} ]
Why \begin{align*}\[Hopf algebra\end{align*} ]? Some natural examples:
\(kG\) the \begin{align*}\[group algebra\end{align*} ].
\(k^G = \mathop{\mathrm{Hom}}_k(kG, k)\) an algebra of functions, forcing distinct group elements to be orthogonal idempotents, take \(\left\{{ P_x {~\mathrel{\Big\vert}~}x\in G }\right\}\) with \(P_x P_y = \delta_{xy} P_y\) ??
Consider category \(\mathsf{H}{\hbox{-}}\mathsf{Mod}^{\mathrm{fd}}\) of finite-dimensional \begin{align*}\[Representation theory\end{align*} ] of \(H\).
Antipode will be invertible when \(H\) is finite dimensional
A lot of structures here: closed under tensors, duals, contains \(k\).
Finite \begin{align*}\[tensor category\end{align*} ] : looks like \(\mathsf{H}{\hbox{-}}\mathsf{Mod}\), \begin{align*}\[Enriched category\end{align*} ] over vector spaces, \begin{align*}\[Monoidal category\end{align*} ], coherent associativity via \begin{align*}\[pentagon axiom\end{align*} ], \begin{align*}\[triangle axiom\end{align*} ].
Define \(\operatorname{Ext} ^n_{\mathsf{C}}(X, Y)\) to be equivalence classes of \(n{\hbox{-}}\)fold extensions, i.e. exact sequences \(0 \to Y \to E_n \to \cdots \to E_1 \to X \to 0\), and \(H^*(\mathsf{C}) \mathrel{\vcenter{:}}= H^*_{\mathsf{C}}(\one, \one) = \bigoplus _{n\geq 0} \operatorname{Ext} ^n_{\mathsf{C}} (\one, \one )\). Can similarly replace \(\one\) with \(X\) to define \(H^*(X)\), which will be a module over \(H^*(\mathsf{C})\).
\begin{align*}\[support variety\end{align*} ] : \(V_{\mathsf{C}}(\one) = \operatorname{mSpec}H^*(\mathsf{C})\), \(V_{\mathsf{C}}(X)\) is a more complicated quotient.
Representation theory of categories: module categories over a category!
Big question: tensor product property. Is there an equality \begin{align*} V_{\mathsf{C}}(X\otimes Y) \overset{?}{=} V_{\mathsf{C}}(X) \cap V_{\mathsf{C}}(Y) .\end{align*}
True for cocommutative \begin{align*}\[Hopf algebra\end{align*} ], some \begin{align*}\[quantum groups\end{align*} ].
Some counterexamples in non-braided monoidal categories. Uses a smash product of modules
See \begin{align*}\[thick ideals\end{align*} ].
Tags: #k_theory #adic #seminar_notes
Reference: Clausen, the K-theory of adic spaces. https://www.youtube.com/watch?v=e_0PTVzViRQ
\begin{align*}\[adic spaces\end{align*} ] : formalism for non-Archimedean geometry.
\begin{align*}\[Formal scheme\end{align*} ] : e.g. formal thickening of a subvariety. Sometimes want to delete a \begin{align*}\[special fiber\end{align*} ].
Definition of ( \begin{align*}\[adic ring\end{align*} ], complete with respect to a finitely-generated ideal, so \(R = \varprojlim R/I^n\)
Some nice features:
Subtleties:
\(\mathsf{Solid}_{\mathbb{Z}}\): abelian bicomplete category of \begin{align*}\[solid mathematics\|solid sets\end{align*} ] Full subcategory of \begin{align*}\[condensed sets\end{align*} ]. Has compact projective generators \(\prod_I {\mathbb{Z}}\)
For morphisms, note \(\mathop{\mathrm{Hom}}( \prod_I {\mathbb{Z}}, {\mathbb{Z}}) = \bigoplus_I {\mathbb{Z}}\)
Let \(\mathsf{Perf}\) be \begin{align*}\[perfect complexes\end{align*} ], why not consider \(K({ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }(X))\) or \(K(\mathsf{Perf}(X))\)?
Need a good category of \begin{align*}\[quasicoherent sheaves\end{align*} ]
What is a \begin{align*}\[presentable category\end{align*} ]?
What is a \begin{align*}\[Tate algebra\end{align*} ]?
What is \begin{align*}\[Arakelov theory\end{align*} ]?
Interesting question in arithmetic statistics: for \(G \in {\mathsf{Grp}}\) finite, how many Galois extensions are there \(K/{\mathbb{Q}}\) with \(G = { \mathsf{Gal}} (K/{\mathbb{Q}})\) and \(\Delta \leq N\) ( \begin{align*}\[discriminant\end{align*} ] for some fixed \(N\)?
Example
One can ask a similar question about \({ \operatorname{Cl}} (K)\) for \(G\in {\mathsf{Ab}}\), or replacing \({\mathbb{Q}}\) with \begin{align*}\[function field\end{align*} ] \({\mathbb{F}}_q(t)\) for \(q=p^n\), and ask questions about frequency of primes \begin{align*}\[ramified primes\end{align*} ], \begin{align*}\[split primes\end{align*} ], or remaining \begin{align*}\[inert primes\end{align*} ].
Cool fact: there is an \begin{align*}\[equivalence of categories\end{align*} ] between finitely-generated extensions \(K/k\) with \(\operatorname{trdeg}(K/k) = 1\) and regular projective \begin{align*}\[curves\end{align*} ] \(C_{/k}\).
\begin{align*}\[Hurwitz spaces\end{align*} ] come up here!
\({\varepsilon}\) is a \(q^i\) \begin{align*}\[Weil number\end{align*} ] if \({\left\lvert { \iota({\varepsilon}) } \right\rvert} = q^{i/2}\) for any embedding \(\iota: { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\hookrightarrow{\mathbb{C}}\).
As a general philosophy, one should expect that \begin{align*}\[moduli space\end{align*} ] problems whose objects have nontrivial automorphisms are representable by \begin{align*}\[stack\end{align*} ], and those without nontrivial automorphisms are \begin{align*}\[representable\end{align*} ] by \begin{align*}\[scheme\end{align*} ].
\begin{align*}\[Weil Conjectures Talks\end{align*} ]
Some old notes from March 10th, 2020
I talked to Erik Schreyer today about some of the research he did with his advisor Jason Cantarella, including his dissertation work (which he spoke about in the Geometry seminar last week) and a few other papers.
His dissertation work involved a cool way to represent arbitrary planar curves by piecewise circular arcs:
From what I understand, this involves fixing a curve (blue), choosing a collection of circles \(C_1, \cdots C_n\) (black) such that each \(C_i\) intersects \(C_{i+1}\) in at least one distinguished point \(p_i\) (pink). The curve traced out by following an arc on \(C_i\) and switching to circle \(C_{i+1}\) at \(p_i\) is intended to yield a good approximation to the original curve, with certain regularity conditions at the \(p_i\) (such as the first derivatives along both arcs agreeing at the point).
Erik’s work actually seems to go a bit farther – he has an algorithm (a curve-closing operator) that actually takes an open curve and produces a closed curve that is nearby in the \(C_1\) norm. He uses this to construct piecewise circular approximations that consist of circles of equal radii, along with some control over the \(C^1\) distance between the original curve and the approximation.
We also talked a bit about another problem Jason was working on, discussed in the following papers:
The symplectic geometry of closed equilateral random walks in 3-space (Cantarella, Shonkwiler 2013)
A Fast Direct Sampling Algorithm for Equilateral Closed Polygons (Cantarella et al 2015)
Dirichlet’s Theorem: An arithmetic progress with \((a, p) = 1\) contains infinitely many primes. As a corollary, one can always find a prime \(q\) that generates \({\mathbb{Z}}_p^{\times}\) for any prime \(p\).
Not every sequence of the form \(0\to A \to A \oplus C \to C \to 0\) splits; take \begin{align*} 0 \to {\mathbb{Z}}\to {\mathbb{Z}}\oplus \bigoplus_{\mathbb{N}}{\mathbb{Z}}/(2) \to \bigoplus_{\mathbb{N}}{\mathbb{Z}}/(2) \to 0 \end{align*} where the first map is multiplication by 2, the second is the quotient map and a right-shift. This can’t split because \((1, 0, \cdots)\) has order 2 in the RHS but pulls back to \((1, 0) \oplus (2{\mathbb{Z}}\oplus 0)\) which has no element of order 2.
See \begin{align*}\[cogroup\end{align*} ].
Reference: Remy van Dobben de Bruyn (Princeton and IAS), “Constructing varieties with prescribed Hodge numbers modulo m in positive characteristic.” Stanford AG Seminar.
One of Serre’s tricks: use group quotients cleverly.
There is a way to directly add/subtract/multiply \begin{align*}\[Hodge diamonds\end{align*} ].
Inverse Hodge problem: when can a Hodge diamond be realized by a smooth \begin{align*}\[projective variety\end{align*} ]? Very hard problem. Want to use this to get information about \(h_{{\mathrm{crys}}}\), i.e. \begin{align*}\[Crystalline cohomology\end{align*} ].
Easier question: look at linear/polynomial relations satisfied by all Hodge diamonds of \({\mathsf{Var}_{/k} }({\mathsf{sm}}, \mathop{\mathrm{proj}})\)?
Main theorems: for a fixed dimension \(n\),
Important tools:
\begin{align*} h(\operatorname{Bl}_2 X) &= h(X) - h(z) + h(E) \\ &= h(X) - h(z) + h(z)(1 + {\mathbb{L}}+ {\mathbb{L}}^2 + \cdots + {\mathbb{L}}^{c-1} )\\ &= h(z) + ({\mathbb{L}}+ {\mathbb{L}}^2 + \cdots + {\mathbb{L}}^{c-1} ) ,\end{align*}
where \(z\) is the point removed and \(E\) is the \begin{align*}\[exceptional divisor\end{align*} ].
Too many primes with \begin{align*}\[supersingular reduction\end{align*} ] implies \begin{align*}\[CM\end{align*} ]. Primes are supersingular about half of the time.
Open image theorems: not known for abelian varieties in general.
\begin{align*}\[Seminars and Talks/2021-04-29 The Galois Action on Symplectic K Theory\end{align*} ]
\begin{align*}\[Seminars and Talks/2021-04-29_2 Yves Andre On the canonical, fpqc and finite topologies\end{align*} ]
\begin{align*}\[Seminars and Talks/2021-04-29_3 Ribet Class groups and Galois representations\end{align*} ]
Some random notes: #todo
-Working out relative homology, an example:
Chain of implications for module properties:
Definitions of common matrix groups:
Good example of exact triangles:
Manifolds from the sheaf perspective, a reference:
Reference: paper on “constructive” algebraic topology J. Rubio, F. Sergeraert / Bull. Sci. math. 126 (2002) 389-412 403
Many constructions in algebraic topology can be organized as solutions of fibration problems.
What are \begin{align*}\[Quillen equivalence\end{align*} ]? These need to preserve the \begin{align*}\[model structure\end{align*} ] on each side presumably.
More fundamental: how should one prove an \begin{align*}\[equivalence of categories\end{align*} ] in general?
Finding \begin{align*}\[adjunction\end{align*} ] is usually easy, because checking isomorphisms on hom sets is concrete.
If you just have a random functor, does it even have right or left adjoints in general? There must be theorems about this. See \begin{align*}\[adjoint functor theorem\end{align*} ].
What is the \begin{align*}\[Stiefel manifold\end{align*} ]?
Description of a certain wrapped \begin{align*}[Fukaya category](#fukaya-category-1)\end{align*} \({\mathcal{O}}\): take the objects to be (Lagrangian) embedded curves, the morphisms are the graded abelian groups \(\hom_{\mathcal{O}}\mathrel{\vcenter{:}}=\qty{\bigoplus_{L_0 \pitchfork L_1} {\mathbb{Z}}/2{\mathbb{Z}}, {{\partial}}}\) where \({{\partial}}\) is given by counting holomorphic strips, localize along small isotopies.
Add to Algebra qual review doc #todo
An ideal \({\mathfrak{p}}\) is prime iff \(JK \subset {\mathfrak{p}}\implies J \subset {\mathfrak{p}}\) or \(K\subset {\mathfrak{p}}\).
A ring is a domain iff the ideal \((0)\) is prime.
Inductively, if \({\mathfrak{p}}\) contains a product of ideals then it contains one of them.
Maximal ideals are prime, since \({\mathfrak{m}}\) maximal implies that \(R/{\mathfrak{m}}\) is a field.
A ring is local iff it has a unique maximal ideal \({\mathfrak{m}}\).
An element \(e\) is idempotent iff \(e^2 = e\).
An \(R{\hbox{-}}\)algebra \(S\) is a ring \(S\) and a homomorphism \(\alpha:R \to S\).
Every ring is a \({\mathbb{Z}}{\hbox{-}}\)algebra in a unique way.
The most interesting commutative algebras are \(S/I\) where \(S = k[x_1, \cdots, x_n]\) for \(k\) a field, \({\mathbb{Z}}\), or the \begin{align*}\[localization\end{align*} ] of a ring at a prime ideal.
See \begin{align*}\[Milnor K theory\end{align*} ]
An appearance of Milnor \(K_2\) in the wild:
How Milnor K-theory shows up in number theory: a conjecture by Tate and Birch:
\begin{align*}\[modular form\end{align*} ] yield 2-dimensional \begin{align*}\[Galois representations\end{align*} ], and there is a classification theorem:
Deligne-Serre Theorem:
Tags: #idle_thoughts
The \begin{align*}\[representation ring\end{align*} ] \(R(G)\): the free \({\mathbb{Z}}{\hbox{-}}\)module on isomorphism classes of irreducible \begin{align*}\[Representation theory\end{align*} ].
Some uses:
Calculate number of \begin{align*}\[rational curve\end{align*} ] in a \begin{align*}\[quintic\end{align*} ] \begin{align*}\[threefold\end{align*} ] (Kontsevich 1995)
Calculate characteristic numbers of a compact \begin{align*}\[homogeneous space\end{align*} ] (Tu 2010)
Derive \begin{align*}\[Gysin formula\end{align*} ] for \begin{align*}\[examples of fiber bundles\end{align*} ] whose fibers are homogeneous spaces (Tu 2011)
Calculate integrals over manifolds as sums over fixed points: \begin{align*}\[Gauss-Bonnet\end{align*} ] and \begin{align*}\[Hopf index theorem\end{align*} ].
What is the \begin{align*}\[Homotopy quotient\end{align*} ]?
If \(G\curvearrowright M\) with \(G\) a compact connected \begin{align*}\[Lie group\end{align*} ], Cartan constructs a chain complex from \(M, {\mathfrak{g}}\).
Is this not precisely the \begin{align*}\[Borel construction\end{align*} ]?
\begin{align*}\[classifying spaces\end{align*} ] : \({\mathbf{B}}S^1 = {\mathbb{CP}}^{\infty}\)
https://www.ams.org/publications/journals/notices/201711/rnoti-p1300.pdf
Manifolds are the place to do differential calculus, \begin{align*}\[scheme\end{align*} ] are the place to do algebra by finding zeros of functions.
\begin{align*}\[Closed point\end{align*} ] : of the form \(V(S) \mathrel{\vcenter{:}}=\left\{{ q\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}q\supseteq S}\right\}\)
http://mathieu.anel.free.fr/mat/doc/Anel-Semiomaths-HomotopyColimit.pdf
Hocolims are \begin{align*}\[infinity groupoids\end{align*} ], equivalently \begin{align*}\[homotopy type\end{align*} ].
There is a functor \(\pi_0: { \underset{\infty}{ \mathsf{Grpd}} }\to {\mathsf{Set}}\).
\begin{align*}\[2021-04-23 Advice on research and problems\end{align*} ]
Setting goals: SMART. Doesn’t work for research though!
\begin{align*}\[attachments/image_2021-04-23-15-54-19.png\end{align*} ]
Make lists, and habitually review/revise/plan.
\begin{align*}\[attachments/image_2021-04-23-15-55-40.png\end{align*} ]
I really like the “keeping a problem list” idea.
Don’t be ashamed to ask people if they have problems you can work on.
Tags: #idle_thoughts
Thinking about the link between group cohomology and homotopy theory: if I have a SES \begin{align*} 0\to A \to B \to C \to 0 ,\end{align*} should one apply a functor like \(K({-}, 1)\)? Is this actually a functor…? We definitely get spaces \(K(A, 1)\) and \(K(B, 1)\), for example, and there must be an induced map between them. Want to make precise what it means to get a SES like this: \begin{align*} 0 \to K(A, 1) \to K(B, 1) \to K(C, 1) \to 0 .\end{align*} One would kind of want this to be part of a \begin{align*}\[fiber sequence\end{align*} ] I guess. But we’re in \({\mathsf{Top}}\) anyway, so there’s no real issue with just doing \begin{align*}\[fibrant and cofibrant objects\end{align*} ],.
Maybe the “right” think to do here is to actually take a classifying \begin{align*}\[groupoid\end{align*} ] (?), which must be some functor like \({\mathbf{B}}: {\mathsf{Grp}}\to {\mathsf{Grpd}}\). Surely this is some known thing. But then what is an “exact sequence of groupoids”…? \begin{align*} 0 \to {\mathbf{B}}A \to {\mathbf{B}}B \to {\mathbf{B}}C \to 0 .\end{align*}
Also, why should such a functor be an exact? It’d kind of be more interesting if it weren’t. Say it’s right-exact, then how might you make sense of \(\mathop{\mathrm{{\mathbb{L} }}}{\mathbf{B}}({-})\)? I think this just needs a model category structure on the source, although it seems reasonable to expect that \({\mathsf{Grpd}}\) would have some simple model structure.
\begin{align*}[l-adic representations](#l-adic-representations-1)\end{align*}
Try computing things like \({ \mathsf{Gal}} ({\mathbb{Q}}( \zeta_3, \sqrt{3})\).
There’s some way to check orders of Galois groups using \begin{align*}\[valuation\end{align*} ]..?
See Néron-Ogg-Shafarevich criterion: \begin{align*}\[good reduction\end{align*} ] iff \begin{align*}\[Inertia\end{align*} ] acts trivially, or \begin{align*}\[semistable reduction\end{align*} ] iff inertia acts \begin{align*}\[unipotently\end{align*} ].
Always have quasi-unipotently, so eigenvalues roots of unity.
\begin{align*}\[Galois representations\end{align*} ] at different primes are related, using local info at a few primes to get global info at all primes.
My main question: does introducing derived stacks somehow make some computation easier?
Integrating over a \begin{align*}\[fundamental class\end{align*} ] :
Appearance of \begin{align*}\[Calabi-Yau\end{align*} ] in \begin{align*}\[Physics\end{align*} ]
Mirror symmetry of CYs:
The major types of “moduli” style invariants
Why care about \begin{align*}\[coherent sheaves\end{align*} ]?
\begin{align*}\[Donaldson-Thomas invariants\end{align*} ] are supposed to relate to \begin{align*}\[Gromov-Witten invariants\end{align*} ] :
Niceness of spaces:
We can’t prove the \begin{align*}\[Tate conjecture\end{align*} ]? I guess this is an arithmetic analog of the \begin{align*}\[Hodge conjecture\end{align*} ]. Serre’s book calls some isomorphism the Tate conjecture and says it’s proved though.
Pithy explanation of a \begin{align*}\[derived scheme\end{align*} ] : a space which can be covered by Zariski opens \(Y\cong \operatorname{Spec}A^*\) where \(A\in {\mathsf{cdga} }_{k}\).
\begin{align*}\[scheme\|schemes\end{align*} ] and \begin{align*}\[stack\|stacks\end{align*} ] can be very singular.
\begin{align*}\[Derived schemes\end{align*} ] and \begin{align*}\[derived stacks\|derived stacks\end{align*} ] act a bit like smooth, nonsingular objects.
Derived modular stacks of \begin{align*}\[quasicoherent sheaf\|quasicoherent sheaves\end{align*} ] over \(X\) remember the entire \begin{align*}\[deformation theory\end{align*} ] of sheaves on \(X\).
#seminar_notes #blog #prisms
One can take \begin{align*}\[etale cohomology\end{align*} ] of varieties, and later refine to schemes, and thus take it for the base field even when it’s not algebraically closed and extract arithmetically interesting information.
\begin{align*}\[prismatic cohomology\end{align*} ], meant to relate a number of other cohomology theories
\begin{align*}\[Prism\end{align*} ] : a pair \((A, I)\) where \(A\) is a commutative ring with a derived Frobenius lift \(\phi:A\to A\), i.e. a \(\delta{\hbox{-}}\)structure.
Fix a scheme and study prisms over it. Need these definitions to have stability under base-change.
Examples:
Prismatic \begin{align*}\[site\end{align*} ] : fix a base prism \((A, I)\) for \(X\) a \(p{\hbox{-}}\)adic \begin{align*}\[formal scheme\end{align*} ] over \(A/I\). Define \begin{align*} (X/A)_\prism = \left\{{ (A, I) \to (B, J) \in \mathop{\mathrm{Mor}}(\mathsf{Prism}), \operatorname{Spf}(B/J) \to X \text{ over } A/I }\right\} ,\end{align*} topologized via the \begin{align*}\[flat topology\end{align*} ] on \(B/J\).
There is a \begin{align*}\[structure sheaf\end{align*} ] \({\mathcal{O}}_\prism\) where \((B, J) \to B\). Take \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma\), which receives a Frobenius action, to define a cohomology theory. Why is this a good idea?
Absolute prismatic sites: for \(X\in {\mathsf{Sch}}(p{\hbox{-}}\text{adic})\), define \begin{align*} X_\prism \mathrel{\vcenter{:}}=\left\{{ (B, J) \in \mathsf{Prism},\, \operatorname{Spf}(B/J) \to X }\right\} .\end{align*} Take \begin{align*}\[sheaf cohomology\end{align*} ] to obtain \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism(X) \mathrel{\vcenter{:}}=\mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_\prism, {\mathcal{O}}_\prism) {\circlearrowleft}_\phi\).
The category \(\mathsf{Prism}\) doesn’t have a \begin{align*}\[final object\end{align*} ], so has interesting cohomology. Relates to \begin{align*}\[Algebraic K theory\end{align*} ] of \({\mathbb{Z}}_p\)?
Questions: let \(X_{/{\mathbb{Z}}_p}\) be a smooth formal scheme.
Bhatt and Lurie: found a stacky way to understand the absolute prismatic site of \({\mathbb{Z}}_p\). Drinfeld found independently.
Construction due to Simpson: take \(X\in {\mathsf{Var}}({\mathsf{Alg}})\), define a de Rham presheaf \begin{align*} X_{\mathrm{dR}}: {\mathsf{Alg}_{\mathbb{C}} }^{\operatorname{fp}} &\to {\mathsf{Set}}\\ R &\mapsto X(R_{ \text{red} }) .\end{align*}
Def: Cartier-Witt stack, a.k.a. the prismatization of \({\mathbb{Z}}_p\)
Plug in a \(p{\hbox{-}}\)nilpotent ring \(R\) to extract all (derived) prism structure on \(W(R)\).
Prisms aren’t base-change compatible without the derived part.
This is a \begin{align*}\[groupoid\end{align*} ].
An explicit presentation: \(\mathsf{WCart}_0(R)\) are distinguished \begin{align*}\[Witt Vectors\end{align*} ] in \(W(R)\). Given by \([a_0, a_1, \cdots ]\) where \(a_0\) is nilpotent and \(a_1\) is a unit. This is a formal affine scheme. \(\mathsf{WCart}= \mathsf{WCart}_0 / W^*\) is a presentation as a \begin{align*}\[stack quotient\end{align*} ].
Start by understanding its points, suffices to evaluate on fields of characteristic \(p\).
If \(k\in \mathsf{Field}(\mathsf{Perf})_{\operatorname{ch. p}}\), \(\mathsf{WCart}(k) = \left\{{ {\operatorname{pt}}}\right\}\), with the point represented by \((W(k), ?)\).
Analogy to understanding \begin{align*}\[Hodge-Tate cohomology\end{align*} ]. Similar easy locus in this stack.
Take 0th component of distinguished \begin{align*}\[Witt Vectors\end{align*} ] to get a diagram
The bottom-left is this \begin{align*}\[Hodge-Tate stack\end{align*} ]
Now has a better chance of being an \begin{align*}\[algebraic stack\end{align*} ] instead of a \begin{align*}\[formal stack\end{align*} ]. Bottom arrow kills the formal direction.
Will be \begin{align*}\[classifying stack\end{align*} ] of a \begin{align*}\[group scheme\end{align*} ] : need to produce a point and take automorphisms.
Take the distinguished element \(V(?) \in W({\mathbb{Z}}_p)\). Produces a map \begin{align*} \operatorname{Spf}({\mathbb{Z}}_p) \xrightarrow{\pi_{\operatorname{HT}}} \mathsf{WCart}^{\operatorname{HT}} .\end{align*}
Upshot: \(\mathsf{WCart}^{\operatorname{HT}}= {\mathbf{B}}W^* [F]\) is a classifying stack. \begin{align*}\[quasicoherent sheaves\end{align*} ] on the left and representations of the (classifying stack of the) group scheme on the right. I.e. \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart}^{\operatorname{HT}}) = \mathop{\mathrm{{\mathbb{R} }}}(W^*[F])\).
Teichmüller lift yields a \({\mathbb{Z}}/p\) grading on the LHS.
Something about Deligne-Illusie? \begin{align*}\[Hodge-to-deRham degeneration\end{align*} ]
Upshot: a \begin{align*}\[divisor\end{align*} ] inside is easy to understand.
Fact: \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart})\) are equivalent to \begin{align*} \varprojlim_{(A, I)\in \mathsf{Prism}} { \mathsf{D} }_{(P, I)-?}(A) .\end{align*}
Diffracted Hodge cohomology: let \(X\in {\mathsf{Schf}}_{{\mathbb{Z}}_p}\). Get a prismatic structure sheaf using the assignment \((A, I) \to \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism \qty{ (X\otimes A/I) / A}\).
Heuristic: \(\operatorname{Spec}{\mathbb{Z}}_p\) should be 1-dimensional over something.
Get an absolute comparison: \(\operatorname{cohdim}\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism (X) \leq d+1\) where \(d = \operatorname{reldim}X_{/{\mathbb{Z}}_p}\).
There is a deRham comparison: \begin{align*} X_{{\mathbb{F}_p}}^* H_\prism(X) \cong \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) .\end{align*}
There is a Hodge-Tate comparison: the object \(H_\prism(X)\) restricted to \(\mathsf{WCart}^{\operatorname{HT}}\) has an increasing filtration with \({\mathsf{gr}\,}_i = \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X, \Omega^i_X)[-i]\).
Combine these comparisons to get \begin{align*}\[Deligne-Illusie\end{align*} ] : if \(\operatorname{reldim}X < p\), then \begin{align*} \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) \cong \bigoplus_{i} \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_{{\mathbb{F}_p}}, \Omega^i[-i]) .\end{align*} Get a lift to characteristic zero, yields Hodge-to-deRham degeneration there.
An \(F{\hbox{-}}\)crystal on \(X_\prism\) is a vector bundle \(\mathcal{E} \in { \mathsf{Vect} }(X_\prism, {\mathcal{O}}_\prism)\)? Plus some extra data.
Infinite tensor product: \begin{align*} I_\prism \otimes F^* I_\prism \otimes(F^2)^* I_\prism \otimes\cdots .\end{align*} Converges to some object \({\mathcal{O}}_\prism \left\{{ 1 }\right\} \in {\operatorname{Pic}}(X_\prism, {\mathcal{O}}_\prism )\), twisted? Yields isomorphism of sheaves after inverting \(I_\prism\), \begin{align*} F^* {\mathcal{O}}_\prism \left\{{ 1 }\right\} \cong I_\prism^{-1}\otimes{\mathcal{O}}_\prism \left\{{ 1 }\right\} .\end{align*}
Some analog of \begin{align*}\[Artin-Schreier\end{align*} ] here, taking fixed points?
There is a natural functor from \(F{\hbox{-}}\)crystals on \(X\) to local \({\mathbb{Z}}_p\) systems on a geometric fiber \(X_?\)?
Main theorem: produces \begin{align*}\[descent\end{align*} ] data, uses work on Beilinson fiber sequence (Benjamin Antieau, Morrow, others?)
Can say \begin{align*} H^i(\Delta_{{\mathbb{Z}}_p}) = \begin{cases} {\mathbb{Z}}_p & i=0 \\ \prod_{{\mathbb{N}}} {\mathbb{Z}}_p & i=1. \end{cases} \end{align*} Can compute using \begin{align*}\[topological Hochschild homology\end{align*} ]? \(\pi_{-1}( {\operatorname{TP}}({\mathbb{Z}}_p) )\) is where the \(i=1\) part comes from.
Prismatic is filtered by things that look like Hodge-Tate
Absolute = arithmetic (take eigenspaces, related to \begin{align*}\[motivic cohomology\end{align*} ], relative = geometric?
Similar to situation in \begin{align*}\[Étale cohomology\end{align*} ] : need absolute and relative to compute either.
Lurie’s Seminar on Algebraic Topology
Lurie’s Topics in Geometric Topology
Some remarks on \({\operatorname{THH}}\) and \(K{\hbox{-}}\)Theory, no clue what the original source was:
\begin{align*}\[Algebraic K theory\end{align*} ] is hard, using \begin{align*}\[topological Hochschild homology\end{align*} ] somehow makes computations easier.
\(K{\hbox{-}}\)theory says something about \begin{align*}\[vector bundles\end{align*} ], \begin{align*}\[topological Hochschild homology\end{align*} ] describes \begin{align*}\[monodromy\end{align*} ] of vector bundles around infinitesimal loops
For \(X\) a nice \begin{align*}\[scheme\end{align*} ], take \(LX\) the derived \begin{align*}\[free loop space\end{align*} ] : the \begin{align*}[derived stacks](#derived-stacks-1)\end{align*} \(\mathop{\mathrm{Maps}}_{ \operatorname{DSt}}(S^1, X)\).
Identify \({\operatorname{THH}}(X) \xrightarrow{\sim} {\mathcal{O}}(LX)\) (global functions)
\begin{align*}\[Dennis trace\end{align*} ] : a comparison \(K(X) \to {\operatorname{THH}}(X)\), takes \(E\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }\) to the canonical monodromy automorphism of the pullback of \(E\) to \(LX\)
Traces are \(S^1{\hbox{-}}\) \begin{align*}\[equivariant\end{align*} ] because loops! Just equip \(K(X)\) with the trivial \(S^1\) action.
Take \begin{align*}\[homotopy fixed points\end{align*} ] to get something smaller than \({\operatorname{THH}}\): \({\operatorname{THC}}^-\), topological negative cyclic homology
Dennis trace is invariant under all covering maps of circles, even multisheeted
Theorem of Dundas-Goodwillie-McCarthy: whenever \(A\to A'\) is a nilpotent extension of connective \begin{align*}\[ring spectra\end{align*} ], \(K(A') \xrightarrow{\sim} K(A) { \underset{\scriptscriptstyle {{\operatorname{TC}}(A)} }{\times} } {\operatorname{TC}}(A')\)
Why should I care about \begin{align*}[derived stacks](#derived-stacks-1)\end{align*} ?
Note from Arun: one can get \begin{align*}\[Topological modular forms\|TMF\end{align*} ] and \begin{align*}\[Topological modular forms\|tmf\end{align*} ] along with their ring structures without doing \begin{align*}\[Obstruction Theory\end{align*} ]
See \begin{align*}\[Arithmetic schemes\end{align*} ]
Idele group for arithmetic schemes
-Actual class group for schemes
Wiesend’s finiteness theorem is one of the strongest and most beautiful results in higher \begin{align*}\[Global class field theory\end{align*} ]?
The main aim of higher \begin{align*}\[Global class field theory\end{align*} ] is to determine the abelian fundamental group \(\pi_1^{{\operatorname{ab}}}(X)\) of a regular arithmetic scheme \(X\), i.e. of a connected \begin{align*}\[regular scheme\end{align*} ] \begin{align*}\[separated scheme\end{align*} ] scheme \begin{align*}\[flat morphism\end{align*} ] and of \begin{align*}\[finite type\end{align*} ] over \({\mathbb{Z}}\), in terms of an arithmetically defined \begin{align*}\[class groups\end{align*} ] \(C(X)\).
Fundamental theorem of class field theory?
#modular_forms
Modular forms can be defined as functions \({\mathbb{H}}\to {\mathbb{C}}\) satisfying weak \(\Gamma{\hbox{-}}\)invariance.
Also sections of a bundle: the \begin{align*}\[modular curve\end{align*} ].
\begin{align*}\[weight of a modular form\end{align*} ] : refers to growth rates of these functions.
Weird fact: \(M_1\) is one-dimensional, but for \(g\geq 2\) we have \(\dim M_g = 3g-3\)
Higher genus generalizations come not from a Teichmüller cover \(T_g \twoheadrightarrow M_g\) or \(M_g\), no one seems to care about those though.
\({\mathbb{H}}_S^g/{\mathsf{Sp}}(2g; {\mathbb{Z}})\) is somehow a model for \begin{align*}\[moduli stack of abelian varieties\end{align*} ], \(M_g\) embeds as a variety since we have the \begin{align*}\[Jacobian\end{align*} ]
\begin{align*}\[Hodge bundle\end{align*} ] : rank \(g\) over \(M_g\), fibers over isomorphism classes are \(H^0(X, K_X)\) where \(K\) is the \begin{align*}\[canonical bundle\end{align*} ], then take the determinant bundle.
#advice
Reminding myself of Ravi’s “Three Things” exercise.
\begin{align*}\[Three Things Exercise\#Process for talks\end{align*} ]
#seminar_notes
What is the \begin{align*}\[universal curve\end{align*} ] over \(M_g\)?
What is the \begin{align*}\[tautological ring\end{align*} ], and how does it relate to the \begin{align*}\[Chow ring\end{align*} ]?
What are the \begin{align*}\[kappa classes\end{align*} ]?
Question: when does \(A^*(M_g) = R^*(M_g)\)?
The integral Chow ring is very unknown. We know it only very recently for \(M_2, \mkern 1.5mu\overline{\mkern-1.5muM_2\mkern-1.5mu}\mkern 1.5mu\). Compare to rational Chow rings: we know them as quotients of polynomial rings for \(M_g, g\leq 23\).
Can stratify \(M_g\) by \begin{align*}\[gonality\end{align*} ].
What are \begin{align*}\[hyperelliptic\end{align*} ] curves?
I definitely have known this at several points in time, yeesh.
Look at \begin{align*}\[Hurwitz space\end{align*} ] : \begin{align*}\[moduli space\end{align*} ] of degree \(n\) covers of \({\mathbb{P}}^1\) by smooth genus \(g\) \begin{align*}\[curves\end{align*} ]? At least this specific one is.
Reference: Chao Li, “Beilinson-Bloch conjecture for unitary Shimura varieties.” Priinceton/IAS NT Seminar
What is the \begin{align*}\[Beilinson Bloch conjecture\end{align*} ]?
Beilinson-Bloch conjecture: generalizes the \begin{align*}\[Birch and Swinnerton-Dyer conjecture\end{align*} ].
What are higher \begin{align*}\[Chow ring\end{align*} ] ? What do they generalize?
What is an \begin{align*}\[adele\end{align*} ]? What is an \begin{align*}\[adele\end{align*} ] point?
I should also review what a \begin{align*}\[place\end{align*} ]really is. Definitely what it means to be an \begin{align*}\[Archimedean place\end{align*} ]. Also double-check the \(v\divides \infty\) notation.
What is an \begin{align*}\[automorphic representation\end{align*} ]?
See \begin{align*}\[Gross-Zagier\end{align*} ] formula.
What is a \begin{align*}\[modular curve\end{align*} ]?
What is a \begin{align*}\[Heegner divisor\end{align*} ] for some \begin{align*}\[imaginary quadratic field\end{align*} ] over \({\mathbb{Q}}\) and why can one use the theory of \begin{align*}\[complex multiplication\end{align*} ] to get it defined over other fields?
Gotta learn \begin{align*}\[modular form\end{align*} ]. They can take values in the complexification of a \begin{align*}\[Mordell-Weil group\end{align*} ]? Also need to know something about \begin{align*}\[Hecke operator\end{align*} ].
What is a \begin{align*}\[Shimura variety\end{align*} ]?
What is a theta series? Something here called an arithmetic theta lift, where some pairing form generalizes Gross-Zagier (?). See Beilinson-Bloch height maybe?
I should read a lot more about \begin{align*}\[Chow ring\|Chow groups\end{align*} ].
What is \begin{align*}\[Betti cohomology\end{align*} ]?
Why is proving that something is \begin{align*}\[modular form\end{align*} ] a big deal?
Look for the Kudla Program in arithmetic geometry, and \begin{align*}\[Kudla-Rapoport conjecture\end{align*} ].
Comment by Peter Sarnak: BSD was first checked numerically for CM elliptic curves!
What is a Siegel \begin{align*}\[Eisenstein series\end{align*} ]? Or even just an Eisenstein series.
See \begin{align*}\[Néron-Tate height\end{align*} ] pairing? Seems like these BB heights can only really be computed locally, then you have to sum over places.
What are the \begin{align*}\[Standard conjectures\end{align*} ]?
Main formula and big theorem:
Seems that we know a lot about the LHS, the right-hand side is new. We don’t know nondegeneracy of the RHS, for example, e.g. the pairing vanishing implying the cycle is zero.
See \begin{align*}\[Tate conjecture\end{align*} ].
Comment from Peter Sarnak: we know very little about where \(L\) functions vanish, except for \(1/2\).
Need to do \begin{align*}\[Resolution of singularities\end{align*} ] when you don’t have a “regular” (integral?) model.
Paper recommended by Juliette Bruce: https://arxiv.org/pdf/2003.02494.pdf
Jonathan Love! Shows some cool consequences of the \begin{align*}\[Beilinson Bloch conjecture\end{align*} ], primarily a 2-parameter family of \begin{align*}\[elliptic curve\end{align*} ] where the image \({\operatorname{CH}}^1(E_1)_0 \otimes{\operatorname{CH}}^1(E_2)_0 \to {\operatorname{CH}}^2(E_1 \times E_2)\) is finite. BB predicts this is always finite when defined over \(k\) a \begin{align*}\[number field\end{align*} ].
I should remind myself what \begin{align*}\[local fields\end{align*} ] and \begin{align*}\[global field\|global fields\end{align*} ] are.
What is the \begin{align*}\[Bloch-Kato\end{align*} ] conjecture? What does it predict for \(L{\hbox{-}}\)functions?
What is the \begin{align*}\[Birch and Swinnerton-Dyer conjecture\end{align*} ]? What does this predict for \begin{align*}\[L function\|L functions\end{align*} ]?
What is a \begin{align*}\[regulator\end{align*} ]?
What is the \begin{align*}\[height pairing\end{align*} ]?
#research_projects
Some motivation for
\begin{align*}\[K3 Surfaces\end{align*}
] : Fermat hypersurfaces \(\sum x_i ^k\) for some fixed \(k\).
Look for \({\mathbb{Q}}{\hbox{-}}\)points, since by homogeneity the denominators can be scaled out to get \({\mathbb{Z}}{\hbox{-}}\)points
\begin{align*}\[Falting's theorem\end{align*} ] : for a \begin{align*}\[curves\end{align*} ] \(C\) with \(g(C) \geq 2\), the number of \begin{align*}\[rational points\end{align*} ] is finite, i.e. \({\sharp}C({\mathbb{Q}}) < \infty\).
Diagonal hypersurfaces \(x_0^k + \cdots + x_n^k = 0\).
\begin{align*}\[Calabi-Yau\end{align*}
] when \(k=n+1\) (maybe a bound instead..?), sharp change in behavior of finiteness of rational points at this threshold.
#seminar_notes
\begin{align*}\[Dehn surgery\end{align*} ] : remove a tubular neighborhood of a knot, i.e. a solid torus, glue back in by some diffeomorphism of the boundary.
\begin{align*}\[L Space conjecture\end{align*} ] simplest \begin{align*}\[Heegard-Floer homology\end{align*} ], rank of \(\operatorname{HF}\) equals cardinality of \(H_{\mathrm{sing}}\).
Left-orderability on groups: a total order compatible with the group operation. Torsion groups can’t be LO: \(x>1 \implies 1 = x^n > \cdots > x > 1\).
\begin{align*}\[taut foliation\end{align*} ] : a geometric condition. Admits a decomposition into leaves where a simple closed curve intersects each transversally?
\begin{align*}\[fibred\end{align*} ] 3-manifolds: take \(\Sigma \times I\) for \(\Sigma\) a surface, glue the top and bottom by some diffeomorphism \(\phi: \Sigma: {\circlearrowleft}\).
Osvath-Szabo: admitting a taut foliation implies being a non-\(L{\hbox{-}}\)space. Is the converse true?
Interesting \begin{align*}\[knot invariants\end{align*} ] : \(\tau, s, g_4(K), \sigma\). Also the Jones, Conway, Alexander polynomials, or even just a coefficient. Note that some of these polynomials can not admit cabling formulas.
Tags: #qual_analysis
\begin{align*}\[quasiisomorphism\end{align*} ] :
See \begin{align*}\[qual review\end{align*} ].
Note from Pete: a common technique on qual Algebra problems is extracting an index 2 subgroup using the Cayley action of \(G\) on itself. These are always normal, so if this exists, \(G\) can’t be simple unless it’s order 2.
The connections between \begin{align*}\[Formal group\end{align*} ] and \begin{align*}\[chromatic homotopy theory\end{align*} ] are key. Read Quillen’s paper, J.F. Adams’ blue book, Ravenel, etc. Ravenel has some slides on Quillen’s work (good entry pt). If you really want to go in depth, I enjoyed learning from Hazewinkel’s book “Formal Groups and Applications.”
#advice #goals
Don’t set goals to be outcome-dependent. Set input goals! E.g. “Write a book that I’m proud of and I like.”
Don’t set goals that depend on factors outside of your control – hitting some metric, getting a specific award, etc.
Check in with your emotional state. It’s not always healthy to push through not feeling like doing something. This can be useful if it’s a matter of discipline, i.e. if pushing through is actually serving you well. But it can also serve you poorly in the long run by exacerbating poor mental state or leading to burnout.
It’s okay to take a break. Check in to ask yourself if it’s coming from a place of self-care or instead procrastination.
Choose to be satisfied! The story you tell yourself after the fact does not change the reality of the past. Does it serve you well to say “I didn’t do enough?” It’s okay to choose to be satisfied.
Have a mental model is the key priority, then go to rote memorization only when absolutely necessary. Remember because you understand the subject!
Use active recall for the learning process – actively test yourself as you’re reading.
See book: “Make It Stick”
The evidence is that popular techniques have low utility: rereading, highlighting, summarizing, taking notes.
Summarizing and taking notes: low utility in general. This has some marginal utility if you’re particularly skilled or trained in summarizing effectively. So not very effective, but do it if it brings you joy!
Active recall and practice testing: high utility! Not very time-intensive, doesn’t require special skills/training.
Do practice testing during each study session. Studies show 10-15% improvement. Try to ask yourself inference questions.
Studies show that those doing retrieval practice actually predict the smallest improvements and achieve the highest improvements. Very counterintuitive, trust the process!
Reading once and practice testing can be more effective than rereading 4 times, and takes less time.
Make first draft of notes with the book closed, no references.
Spaced repetition: also works on a micro scale! E.g. within a single study session.
Practice a little bit each day over a long period of time.
Extremely important: scope your subject. Know the broad outline of the course inside and out. Can scope based on actual exams.
Start from topics you don’t know as well. E.g. work backwards though lecture notes.
Try interleaved practice: don’t necessarily need to master a topic before moving on, realizing that you’ll be reviewing it several times again before it’s needed.
Idea for tracking system: put course outline by topics into a spreadsheet, then record dates of review next to each topic. Color code cells based on quality of recall.
Revision needs to be a fluid process, not worth timetabling down to each section due to different time requirements for each. Pick what makes your brain work the hardest.
Reference: Link to PDF
See \begin{align*}\[Spectral sequence\end{align*} ].
What is a \begin{align*}\[Dirichlet character\end{align*} ]?
What is a \begin{align*}\[Dirichlet L function\end{align*} ]?
\begin{align*} L(s ; \chi):=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}} =\prod_{p} \qty{ 1-\chi(p) p^{-s} }^{-1} .\end{align*}
How is a \begin{align*}\[Bernoulii number\end{align*} ] defined? Generalized Bernoulli numbers:
What is the \begin{align*}\[conductor\end{align*} ] of a \begin{align*}\[Dirichlet character\end{align*} ]?
What is the \begin{align*}\[J-homomorphism\end{align*} ]?
How is it defined in terms of \begin{align*}\[loop space\end{align*} ]?
How is it defined in terms of \begin{align*}\[framed cobordism\end{align*} ]? What is a \begin{align*}\[framed\|framing\end{align*} ]?
How is it defined in terms of \begin{align*}\[Thom space\end{align*} ]? What is a Thom space?
What is a \begin{align*}\[complex oriented cohomology theory\end{align*} ]?
What is a \begin{align*}\[uniformizer\end{align*} ]?
What is a \begin{align*}\[Kan extension\end{align*} ]? #todo
Kan extensions:
What are \begin{align*}\[derived functors\end{align*} ] in a \begin{align*}\[Model category theory\|model category\end{align*} ]?
What is a \begin{align*}\[Quillen adjunction\end{align*} ]?
What is a \begin{align*}\[Quillen equivalence\end{align*} ]?
What is an \begin{align*}\[infinity categories\|infinity category\end{align*} ]?
What is a \begin{align*}\[topological category\end{align*} ]?
What does it mean for a category to be \begin{align*}\[Enriched category\|enriched\end{align*} ]?
What is the \begin{align*}\[homotopy category\end{align*} ] of a \begin{align*}\[Model category theory\|model category\end{align*} ]?
What is the model structure on (differential graded) vector spaces? What are the fibrations, cofibrations, and weak equivalences?
What does an augmented algebra correspond to a pointed affine \begin{align*}\[scheme\end{align*} ]?
How are initial and final objects defined in \begin{align*}\[infinity categories\|infinity category\end{align*} ]?
#rational_points
\begin{align*}\[Vojta's conjecture\end{align*} ] : Gives good \begin{align*}\[height\end{align*} ] estimates, can be used to prove something is finite.
\begin{align*}\[Northcott's theorem\end{align*} ] Used in arithmetic dynamics since bounded height and bounded degree implies finite.