1 2021-11-22

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1.1 01:01

• You can define \(p{\hbox{-}}\)curvature for arithmetic schemes. Are there analogs of the Riemann curvature tensor? Ricci curvature?
  • How to make Floer or Morse homology work for varieties or schemes? Not clear how to define things like gradient flows algebraically.

1.2 UGA Topology Seminar, Irving Dai, Equivariant Concordance and Knot Floer Homology

• WIP! Joint with Mallick and Stoffregen.
• Equivariant knots: pairs \((K, \tau)\) where \(K \subseteq S^3\) and \(\tau: S^3{\circlearrowleft}\) an orientation-preserving involution preserving \(K\), so \(\tau(K) = K\).
• Symmetries of the trefoil:

• 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

  • Equivariant concordance invariants
  • Functions of these bound \(\tilde g_4(K)\) from below.

• 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.

  • JMZ: higher genus construction using knot floer homology. Needs high genus, won’t work for slice discs.
  • H, HS: for slice discs using Khovanov homology.

• 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

  • \(W \in \operatorname{ZHB}^4\) with \({{\partial}}W = S^3\)
  • \(\tau_W: W\to W\) extends \(\tau\) (in any way!)
  • \(\tau_W \Sigma \isotopic \Sigma\) rel \(K\).

• 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.

  • Q: given a number of arrows, can this be achieved by picking a suitable genus?

• 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.

  • So this can show non-isotopic, and require many stabilizations to become isotopic.

• 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 \(p\)!

1.3 19:57

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*} ]

2 2021-11-10

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2.1 16:20

Hector Pasten, UGA NT seminar.

• Mordell’s conjecture: for \(C\) a curve, \(C({\mathbb{Q}}) < \infty\).
  • Chabauty: if \({\operatorname{rank}}J({\mathbb{Q}}) > g = \dim J({\mathbb{Q}})\), then \(C({\mathbb{Q}})\) is finite.
  • Faltings: Proof using heights on moduli spaces
  • Vojta: Proof by Diophantine approximation.
• Abel-Jacobi map: \(C\to J_C\) by \(x \mapsto [x-x_0]\).
• Chabauty’s proof: let \(\Gamma\) be the \(p{\hbox{-}}\)adic closure of \(J({\mathbb{Q}})\) in \(J({ {\mathbb{Q}}_p })\), which is a \(p{\hbox{-}}\)adic Lie subgroup of \(J({ {\mathbb{Q}}_p })\). Interpret \(\Gamma \cap C({ {\mathbb{Q}}_p })\) as zero loci of \(p{\hbox{-}}\)adic analytic functions of \(C({ {\mathbb{Q}}_p })\), constructed using integration.
• See \begin{align*}\[good reduction\end{align*} ], \begin{align*}\[hyperplane section\end{align*} ].
• Nice: smooth, projective, \begin{align*}\[geometrically irreducible\end{align*} ].
• Looks hyperbolic: contains no elliptic curves.
• First Chern number: self-intersection of the canonical divisor.
• Reduction map \({ \text{red} }: A({ {\mathbb{Q}}_p }) \to A({\mathbb{F}}_p)\), take residue discs \(U_x \mathrel{\vcenter{:}}={ \text{red} }^{-1}(x)\) for \(x\in A({\mathbb{F}}_p)\). Bound the number of points in \(X({\mathbb{Q}}_p) \cap\Gamma \cap U_x\).
• Fat point: \({ {\mathbb{Q}}_p }[z]/\left\langle{z^n}\right\rangle\) for some \(n> 1\).

3 2021-11-09

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3.1 00:11

3.2 15:51

• 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.

  • The pair \((J(X), \Theta)\) determines a cubic threefold, where \(\Theta\) is the theta divisor, which has a unique singular point.

• 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)).

4 2021-11-08

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4.1 15:05

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.

  • Note that lens spaces have this property!

• 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 \(S^1\times S^2\).

5 2021-11-05

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5.1 01:31

Differential forms on (derived) \begin{align*}\[stacks\end{align*} ]:

What is non-commutative geometry?

Category of singularities:

Bloch’s conductor conjecture: