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