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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.

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 \(p\)!

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*} ]

2021-11-10

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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 \(U_x \coloneqq{ \text{red} }^{-1}(x)\) 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 \(n> 1\).

2021-11-09

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00:11

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)).

2021-11-08

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Refs: ?

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 \(S^1\times S^2\).

2021-11-05

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01:31

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

What is non-commutative geometry?

Category of singularities:

Bloch’s conductor conjecture:

Other Stuff

On harmonic bundles:

13:41

Existence of spin and string structures: kind of like applying a functor to the Whitehead tower and asking for sections of the image tower:

Link to Diagram

2021-11-04

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01:36

2021-11-03

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15:09

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 \(X\)”.

  • Define GIT quotient as \(X{ \mathbin{/\mkern-6mu/}}G\coloneqq\mathop{\mathrm{proj}}\bigoplus _{i\geq 0} H^*(X; {\mathcal{L}}{ {}^{ \scriptscriptstyle\otimes_{k}^{i} } } )^G\), where by Hilbert if \(G\) is reductive then the invariants are finitely generated.

• 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.

  • Kobayashi conjecture (1970): any generic hypersurface \(X \subseteq {\mathbb{P}}^{n+1}\) of degree \(d_n \gg 1\) is Brody hyperbolic.
  • Griffiths-Lang conjecture (1979): any projective variety \(X\) of general type is weakly Brody hyperbolic.
  • Theorem, Riedl-Young 2018: if for all \(n\) there exists a \(d_n\) such that GL holds for generic hypersurfaces of degree \(d\geq d_n\), then the Kobayashi conjecture is true for them.

• \(\widehat{U}\) 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.

16:23

• 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

19:43

Idk I just like this:

2021-11-01

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15:03 UGA Topology Seminar

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 \(\alpha \vee d\alpha > 0\).

  • Example: \(\xi = \ker(\,dz+ r^2 \,d\theta)\) on \({\mathbb{R}}^3\).

• A transverse knot is a knot positively transverse to \({ \left.{{ \alpha }} \right|_{{K}} } > 0\).

  • Knots braided about the \(z{\hbox{-}}\)axis are naturally transverse, so study transverse knots via braids.

• 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 \(Y_{\phi} \coloneqq S\times I / (x,1) \sim (\phi(x), 0)\).

• 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.

2021-10-29

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21:10

https://arxiv.org/pdf/1904.06756.pdf

Some notes on \begin{align*}\[quadratic differentials\end{align*} ]:

• See \begin{align*}\[central charge\end{align*} ], \begin{align*}\[stability conditions\end{align*} ] on a \begin{align*}\[triangulated category\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:

• What is the \begin{align*}\[divisor associated to a section\end{align*} ]? Answered here:

• 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 \({ \operatorname{Cl}} (X)\).

• Computations of the cohomology of the trivial and canonical bundles:

2021-10-27

Tags: #quick_notes

Refs: ?

15:17

Kristin DeVleming, UGA AG seminar talk on moduli of quartic \begin{align*}\[K3 surfaces\end{align*} ].

• See \begin{align*}\[K-stability\end{align*} ], \begin{align*}\[log Fano pairs\end{align*} ], \begin{align*}\[Fano varieties\end{align*} ], \begin{align*}\[Hassett-Keel program\end{align*} ]
  • There’s a way to take the volume of the anticanonical divisor \(-K_X\), see the \begin{align*}\[delta invariant\end{align*} ].
  • Defines a moduli space with a natural \begin{align*}\[wall crossing\end{align*} ] framework.
  • See \begin{align*}\[du Val singularities\end{align*} ] and \begin{align*}\[ADE singularities\end{align*} ].
  • A \begin{align*}\[polarized K3\end{align*} ] is a pair \((S, L)\) with \(S\) a \begin{align*}\[K3\end{align*} ] and \(L\) an ample line bundle.
  • From a \begin{align*}\[Hodge theory\end{align*} ] perspective, there is a natural \begin{align*}\[period domain\end{align*} ].
  • See \begin{align*}\[GIT\end{align*} ] \begin{align*}\[moduli spaces\end{align*} ], \begin{align*}\[Hodge bundle\end{align*} ], \begin{align*}\[Heegner divisor\end{align*} ]
  • There is a map \(\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{M}}\mkern-1.5mu}\mkern 1.5mu^{\mathrm{GIT}} \to {\mathcal{F}}_4^*\) where the LHS are quartics in \({\mathbb{P}}^1\), and the RHS has two nontrivial divisors \(H_k\) parameterizing \begin{align*}\[hyperellptic\end{align*} ] K3s and \(H_u\) parameterizing \begin{align*}\[unigonal\end{align*} ] K3s.
  • See \begin{align*}\[weighted projective space\end{align*} ], here \({\mathbb{P}}^1 \times {\mathbb{P}}^1\) is a smooth quadric in \({\mathbb{P}}^3\) while a singular one is \({\mathbb{P}}(1, 1, 2)\):

• Can reduce moduli of K3s to studying moduli of curves plus stability conditions. Studying unigonal K3s reduces to studying \begin{align*}\[elliptic fibrations\end{align*} ], i.e. maps \(S\to C \subseteq {\mathbb{P}}^3\) a \begin{align*}\[twisted cubic\end{align*} ] whose fibers are elliptic.
• By Leza-O’Grady, there is a nice VGIT wall crossing framework.
• Theorem: can interpolate between \({\mathcal{M}}_4^{\GIT}\) and \({\mathcal{F}}_4^*\) via a sequence of explicit \(K{\hbox{-}}\)moduli \begin{align*}\[wall crossings\end{align*} ]
  • #idle_thoughts Sequences of wall crossings look like \begin{align*}\[correspondences\end{align*} ] or \begin{align*}\[spans\end{align*} ]

16:24

Jiuya Wang’s, UGA NT seminar talk

• See \begin{align*}\[ramified primes\end{align*} ], \begin{align*}\[inertia group\end{align*} ], \begin{align*}\[class group\end{align*} ], \begin{align*}\[discriminant\end{align*} ]
• \begin{align*}\[Malle's conjecture\end{align*} ] implies the \begin{align*}\[inverse Galois problem\end{align*} ].
• \begin{align*}\[Kronecker-Webber theorem\end{align*} ] in \begin{align*}\[class field theory\end{align*} ].

2021-10-25

Tags: #quick_notes

Refs: \begin{align*}\[Advice\end{align*} ]

00:00

• #advice Don’t put off problem-solving until another day! Work on something small today rather than waiting for “enough” time to devote some grand problem. Part of the trick is constant progress and assembling small results into large ones (and conversely breaking large problems into such small pieces)

16:16

• 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.

  • Example: \(U({\mathfrak{g}})\wedge k[G]\).

• 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: \(\cmods{H} { { \, \xrightarrow{\sim}\, }}{{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]}{\hbox{-}}\mathsf{coMod}}{ { \, \xrightarrow{\sim}\, }}{{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]^{\sigma}}{\hbox{-}}\mathsf{coMod}}\).

2021-10-24

Tags: #quick_notes

Refs: ?

00:01

• What is a \begin{align*}\[quiver\end{align*} ]?
• What is a \begin{align*}\[K3\end{align*} ]?
• What is \begin{align*}\[GIT\end{align*} ]?
• What is \begin{align*}\[wall-crossing\end{align*} ]?

2021-10-19

23:01

• 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 \(d^p = 0\)?

  • c/o Ivo https://arxiv.org/pdf/1205.1814.pdf

• \begin{align*}\[mock modular forms\end{align*} ], \begin{align*}\[moonshine\end{align*} ], \begin{align*}\[elliptic cohomology\end{align*} ]

23:12

2021-10-18

15:07

Tags: #knots #concordance #geometric_topology

2021-10-13

00:18

• A bit on \begin{align*}\[lambda sequences\end{align*} ] :
  • Let \(\Lambda\) be the category of based \begin{align*}\[Finset|finite sets\end{align*} ] \(\mathbf{n}=\{0,1,2, \cdots, n\}\) with base point 0 and based injections. The morphisms of \(\Lambda\) are generated by permutations and the ordered injections \(s_{i}^{k}: \mathbf{k}-\mathbf{1} \rightarrow \mathbf{k}\) that skip \(i\) for \(1 \leq i \leq k .\) It is a \begin{align*}\[symmetric monoidal category\end{align*} ] with wedge sum as the symmetric monoidal product. Let \((\mathscr{V}, \otimes, \mathcal{I})\) be a bicomplete symmetric monoidal category with initial object \(\varnothing\), terminal object \(*\).

Tags: #quick_notes

2021-10-08

21:03

• \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:

  • \({\mathsf{QCoh}}(\operatorname{Spec}R) { { \, \xrightarrow{\sim}\, }}{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)
  • Use \(p: \operatorname{Spec}R\to {\mathbf{B}}G\) induced by \(G\to R\) to induce a pullback functor \(p^*: {\mathsf{QCoh}}({\mathbf{B}}G)\to {\mathsf{QCoh}}(\operatorname{Spec}R) \cong {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).
  • Set up a adjunction that yields a comonad equivalent to \(F: ({-})\otimes_R \Gamma(G)\).
  • Apply \begin{align*}\[Barr-Beck\end{align*} ] :
    • Given an adjunction \(\adjunction{L}{R}{\mathsf{D}}{\mathsf{C}}\), get a comonad \(LR\in [\mathsf{C}, \mathsf{C}]\).
    • Then every \(X\in \mathsf{D}\) yields \(L(X) \in {{LR}{\hbox{-}}\mathsf{coMod}}\), and \(\mathsf{D} \xrightarrow{L} \mathsf{C}\) factors:

Link to diagram

• \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} \coloneqq\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \coloneqq\mathsf{C} .\end{align*} Then \(LR \coloneqq p^*p_*\), 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 \(\mathsf{C} \coloneqq{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), and \(F\in [\mathsf{C}, \mathsf{C}]\) to be \(F({-}) \coloneqq({-})\otimes_R \Gamma(G)\). 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 \(p^*p_*\).

22:52

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*} ]

• Vector bundles as descent data: consider describing \(E\to X\); one needs the \begin{align*}\[cocycle condition\end{align*} ]. This means choosing \({\mathcal{U}}\rightrightarrows X\) and bundle automorphisms \(\phi_{ij}: (U_i \cap U_j)\times {\mathbb{R}}^n {\circlearrowleft}\) of the trivial bundle.
  • We then want to glue up to obtain some \(E\) over \(X\): finding local bundle isomorphisms \(\phi_i: U_i \times {\mathbb{R}}^n { { \, \xrightarrow{\sim}\, }}{ \left.{{E}} \right|_{{U_i}} }\) with \(\phi_{ij} = \phi_i \circ \phi_j^{-1}\) on \(U_i \cap U_j\). The cocycle condition is necessary, and for topological vector bundles, also sufficient.
  • How to glue: set \(E \coloneqq{\textstyle\coprod}_{i} (U_i \times {\mathbb{R}}^n)/\sim\) where \((x, \mathbf{v})\sim (x, \phi_{ij}(\mathbf{v}))\) with the quotient topology.
• Alternative formulation:
  • Let \({\mathcal{U}}\rightrightarrows X\) and define \(Y\coloneqq\displaystyle\coprod_i U_i\), which induces \(Y \xrightarrow{\pi} X\) by the inclusions \(U_i \hookrightarrow X\).
  • Then \begin{align*} Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } } = \displaystyle\coprod_{(i, j)\in I{ {}^{ \scriptscriptstyle\times^{2} } }} U_i \cap U_j .\end{align*} The cocycle condition becomes the existence of an isomorphism of bundles over \(Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }\):

Link to diagram

• Note that pullbacks of trivial bundles are trivial, so this is an automorphism of the trivial bundle on \(Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }\)
• The cocycle condition becomes an identity among bundle isomorphisms on \(Y{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{3} } }\): \begin{align*} p_{12}^* \phi \circ p_{23}^* \phi = p_{13}^*\phi \end{align*} as maps \(p_3^*\tilde E\to p_1^* \tilde E\). Local trivializations translate to \(\pi^* E \cong \tilde E\), the trivial bundle.

23:25

• 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.

  • The forward map is \(V\mapsto (V\otimes_{\mathbb{R}}{\mathbb{C}}, f)\) with \(f\) the generator \(f\in { \mathsf{Gal}} ({\mathbb{C}}_{/ {{\mathbb{R}}}} )\), and the inverse is \((V, f)\mapsto V^f\), the \(f{\hbox{-}}\)invariant subspace.

• 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 \(\operatorname{Spec}R \coloneqq\mathsf{CRing}(R, {-})\), 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 \(i\).

  • This recovers \({\mathbb{P}}^1_{{\mathbb{Z}}}({\mathbb{C}}) = {\mathbb{P}}^1_{/ {{\mathbb{C}}}}\) classically, since sub-vector spaces are direct summands.
  • \({\mathbb{P}}^1_{\mathbb{Z}}({\mathbb{Z}}[t])\) induces a continuously varying family of 1-dimensional subspaces of \({\mathbb{C}}^2\)? Somehow, even though \({\mathbb{C}}\) isn’t in the definition..

• 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 \(s\otimes(rm) \coloneqq\alpha(r) (s\otimes m) = (\alpha(r)s)\otimes m\).

• 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.

  • Free implies flat, and every module over a field is free.

• \(S\) 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 \({\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)\),

  • \begin{align*}\[Descent data\end{align*} ] : pairs \((M, \phi)\) where \(M\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}}\) and \(\phi: M\otimes_RS { { \, \xrightarrow{\sim}\, }}S\otimes_R M\) is a twist isomorphism.

• 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 \(\alpha\in [G, LF]\).

  • Example: \(G:{\mathsf{A}{\hbox{-}}\mathsf{Mod}}\to \mathsf{A}\) into some \begin{align*}\[abelian category\end{align*} ]. Here \begin{align*}\[simplicial resolution\end{align*} ] by projective objects for projective resolutions, and \({\mathbb{L}}G\) is the left \begin{align*}\[Kan extension\end{align*} ] of \(G:\mathsf{C} \to K^-(\mathsf{A})\) along the inclusion \(\mathsf{C} \to K^-(\mathsf{A})\), where \(\mathsf{C} \leq K^-(\mathsf{A})\) are complexes of projective modules. So this replaces \begin{align*}\[cofibrant replacement\end{align*} ].

Tags: #quick_notes

2021-10-06

00:22

Tags: #terms_and_questions

• What is a \begin{align*}\[Higgs bundle\end{align*} ]?

Tags: #quick_notes

2021-10-05

DAG-X

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*} ]

10:49

Weak weak approximation would imply a positive answer to the \begin{align*}\[inverse Galois problem\end{align*} ].

20:02

\begin{align*}\[attachments/2021-10-05_20-02-50.png\end{align*} ]

Elliptic Cohomology Paper

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*} ]

22:49

• 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*} ]

Volcano Stuff

• How \begin{align*}\[K-Theory\end{align*} ] goes:

  • Form a \begin{align*}\[symmetric monoidal category\end{align*} ] \(\mathsf{C}\), which is a commutative monoid object in \begin{align*}\[infinity categories\end{align*} ].
  • Apply \({ \mathsf{core} }: \mathsf{Cat}\to{\mathsf{Grpd}}\) to replace \(\mathsf{C}\) with \({ \mathsf{core} }\mathsf{C}\), which separates isomorphism classes into separate connected components. It turns out this lands in \({\mathbb{E}}_\infty{\hbox{-}}\)spaces, i.e. commutative monoid objects in infinity-groupoids.
  • Apply \begin{align*}\[group completion\end{align*} ] of \((\infty, 1){\hbox{-}}\)categories to get an abelian group object in infinity-groupoids.
  • Identify these with connective \begin{align*}\[spectra\end{align*} ].
  • Include into the category of all spectra.

• Minor aside: \({\mathbf{B}}\mathsf{C} \coloneqq{ {\left\lvert {{ \mathcal{N}({\mathsf{C}}) }} \right\rvert} }\).

• 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..?

  • Then take covering category: objects are based surjections \(E_1 \twoheadrightarrow E_2\), morphisms
  • Restrict to “covering spaces”: fibers are finite and discrete.

2021-10-04

01:02

Refs: \begin{align*}\[algebra valued differential forms\end{align*} ]

• https://www.wikiwand.com/en/Vector-valued_differential_form

• https://www.wikiwand.com/en/Lie_algebra-valued_differential_form

• https://www.wikiwand.com/en/Adjoint_bundle

• \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*} ] :

  • 

2021-10-03

• \begin{align*}\[perfect complexes|perfect complex\end{align*} ] and perfect modules?

Spectra Stuff

Tags: #stable_homotopy

Producing a LES:

• Take a map \(A \xrightarrow{f} B\)
• Extract cofibers to get \(A \to B \to {\operatorname{hocofib}}(f) \to \cdots\)
• Apply \([\mathop{\mathrm{{\Sigma_+^\infty}}}({-}), E]_{-n}\)

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*}

• \begin{align*}\[Cohomology operations\end{align*} ] : natural transformations \(E^n({-})\to F^m({-})\).
• Classified by maps \(E_n \to F_m\), i.e. \(F^m(E_n)\).
• E.g. \begin{align*}\[Steenrod squares\end{align*} ] \(\operatorname{Sq}^i \in [K(C_2, n), K(C_2, n+i)]\).
• They’re in fact stable, so live in \(HC_2^*(HC_2)\).
• In general, algebras of stable operations for a cohomology theory \(E\) are exactly \(E^*(E)\).

Categories

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:

Link to Diagram

• \begin{align*}\[extranatural transformations|Extranatural transformations\end{align*} ] are given by a certain string calculus:

• 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 \(F(f)(G(x)) \subseteq G(y)\).

Lie Algebras?

References: https://arxiv.org/pdf/0801.3480.pdf and https://people.math.umass.edu/~gwilliam/thesis.pdf

Tags: #reading_notes #lie_algebras

• Some \begin{align*}\[spin\end{align*} ] stuff and \begin{align*}\[algebra valued differential forms\end{align*} ]

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:

Link to Diagram

• Defining \begin{align*}\[factorization algebras\end{align*} ] :

2021-10-02

00:20

Tags: #idle_thoughts

Idk this weird thing

Link to diagram

2021-09-24

14:33

Tags: #terms_and_questions #unanswered_questions

• Can you use the \begin{align*}\[Algebraic de Rham complex\end{align*} ] to get a volume form?
• Can you integrate it over the space?
• Can one define a notion of volume for \({\mathbb{A}}^n_{/ {k}}\) with \(k\) an arbitrary field?

2021-09-23

22:37

Tags: #terms_and_questions

• \begin{align*}\[Hilbert symbol\end{align*} ]
• \begin{align*}\[Norm-residue symbol\end{align*} ]

2021-09-20

01:29

https://math.stanford.edu/~conrad/papers/hypercover.pdf

• See \begin{align*}\[simplicial object\end{align*} ] :

2021-09-19

22:51

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:

Link to Diagram

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!!

2021-09-16

20:04

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.

  • Arrows in \(\mathsf{C}(x, y)(u\to v)\) are \begin{align*}\[deformations\end{align*} ] of \(u\) to \(v\).
  • 2-functors are enriched functors.

• \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 \({\mathbf{B}}C \coloneqq{ {\left\lvert {{ \mathcal{N}({C}) }} \right\rvert} }\).

• 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*} ] :

• Any \begin{align*}\[monoidal category\end{align*} ] is a \begin{align*}\[2-category\end{align*} ] with one object.

2021-09-14

14:45

Tags: #representation_theory #terms_and_questions #number_theory #langlands

• \begin{align*}\[Cartan\end{align*} ] subgroup

  • Centralizer of a maximal torus.

• \begin{align*}\[Borel\end{align*} ] subgroup

  • Maximal connected solvable subgroup
  • Why care: critical to structure theory of simple \begin{align*}\[reductive algebraic group\end{align*} ]. Uses pairs \((B, N)\) where \(N = N_G(T)\) is the normalizer of a maximal torus.

• \begin{align*}\[Parabolic\end{align*} ] subgroup

  • Literally any \(P\leq G\) such that \(B \subseteq P \subseteq G\)
  • \(G/P\) is a complete variety, so all projections \(X\times ({-}) \to ({-})\) are closed maps.
  • \(G/B\) is the largest complete variety since \(B \subseteq P\) for all \(P\).

• \begin{align*}\[local fields\end{align*} ]

  • Complete with respect to a topology induced by \(v\) a discrete valuation with \(\kappa\) finite.

• \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*}\[Value group\end{align*} ] : \(\operatorname{im}v\)
  • \begin{align*}\[Valuation ring\end{align*} ] : \(R_v \coloneqq\left\{{v(x) \geq 0}\right\}\)
  • Prime/maximal ideal: \({\mathfrak{m}}_v \coloneqq\left\{{v(x)>0}\right\}\)
  • Residue field \(\kappa_v \coloneqq R_v/{\mathfrak{m}}_v\)
  • \begin{align*}\[place|places\end{align*} ] : \(\left\{{v}\right\}/\sim\) where \(v_2\sim v_1 \iff v_2 = \phi \circ v_1\).
  • \begin{align*}\[uniformizer\end{align*} ] : for \(R\) a DVR, a generator \(\pi\) for the unique maximal ideal, so \(R^{\times}\left\langle{\pi}\right\rangle = R\) and \(x\in R \implies x = u\pi^k\)

• \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 \({\mathbb{F}}_q { \left( {(} \right) } t))\).

  • For a 1-dim variety: \(\operatorname{ff}k[X]\), the fraction field of the coordinate ring.

• 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*} ]

  • Existence of infinitesimals, i.e. for a \({\mathbb{Z}}{\hbox{-}}\)module with a linear order, \(x\) is infinitesimal with respect to \(y\) if \(nx < y\) for all \(n\)
  • E.g. \({\mathbb{R}} { \left( {(} \right) } x)\) or \({\mathbb{Q}} { \left( {(} \right) } x)\), \(1/x\) is infinitesimal. Or \({ {\mathbb{Q}}_p }\).
  • Nonarchimedean local fields are totally disconnected.

• \begin{align*}\[proper morphism\end{align*} ]

  • Separated, finite type, universally closed (so for \(X\to Y\), all projections \(X{ \underset{\scriptscriptstyle {Y} }{\times} }Z\to Z)\) are closed maps).
  • For spaces: preimages of compact subspaces are compact.
  • For locally compact Hausdorff spaces: continuous and closed with compact fibers.

• \begin{align*}\[Iwahori\end{align*} ] subgroup

  • Subgroup of an algebraic group over a nonarchimedean local field, analogous to a Borel.

• 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.

  • Motivates defining \(K_\infty \coloneqq\colim_n L(\mu_{p^{n+1}})\), using \({ \mathsf{Gal}} (K_n{}_{/ {K}} ) = C_{p^n}\) so \(G\coloneqq{ \mathsf{Gal}} (K_\infty {}_{/ {K}} ) = { {\mathbb{Z}}_p }\). Set \(I_n = { \operatorname{cl}} (K_n)[p]\) to be the \(p{\hbox{-}}\)torsion in the ideal class group of \(K_n\), form \(I\coloneqq\colim_n I_n\) using norm maps to get module structure, recover info about \({ \operatorname{cl}} (K)[p]\).

• 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.

  • One defining method: interpolate special values.

• Actual definition of Dirichlet characters:

• Fundamental lemma in \begin{align*}\[The Langlands Program\end{align*} ]
  • Relates orbital integrals on a \begin{align*}\[reductive group\end{align*} ] over a \begin{align*}\[local fields\end{align*} ], to “stable” orbital integrals on its endoscopic groups.
    • Endoscope: \(H\leq G\) a quasi-split group whose Langlands dual \(H {}^{ \vee }\) is the connected component of \(C_{G {}^{ \vee }}(x)\) for \(x\in G {}^{ \vee }\) some semisimple element.
  • Want to get at \begin{align*}\[automorphic forms\end{align*} ] and the arithmetic of \begin{align*}\[Shimura varieties\end{align*} ]
  • Some “stabilized” version of the \begin{align*}\[Grothendieck-Lefschetz Trace Formula\end{align*} ]?

22:17

• 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*} ]

  • For \(f \in \mathop{\mathrm{Mor}}_{\mathsf{Sch}}(X, Y)\) finite type and \(X, Y\) locally Noetherian, \(f\) is etale at \(y\in Y\) if \(f^*: {\mathcal{O}}_{f(y)} \to {\mathcal{O}}_y\) is flat and \({\mathcal{O}}_{f(y)}/{\mathfrak{m}}_{f(y)} \to {\mathcal{O}}_{f(y)}/ f^*({\mathfrak{m}}_{f(y)} {\mathcal{O}}_y)\) is a finite separable extension.

• Central extension

• Fiber functor

• Algebraic fundamental group

  attachments/2021-09-14_22-29-30.png❗

• 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

2021-09-12

Tags: #homological_stability #representation_theory

\({\mathsf{FI}}{\hbox{-}}\)modules (23:45)

Reference: Church-Ellenberg-Farb

• #open_problems : what are the \begin{align*}\[characters\end{align*} ] of representations for \(S_n\) acting on certain vector spaces:
  • \(H^*({\mathrm{Conf}}_n(X))\)
  • \(H^*({ \mathcal{M}_{g} })\) and its tautological ring \(R^*({ \mathcal{M}_{g} })\)

    • 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.

• \({\mathsf{{\mathsf{FI}}}{\hbox{-}}\mathsf{Mod}} \coloneqq F\in {\mathsf{Fun}}({\mathsf{FI}}, {\mathsf{k}{\hbox{-}}\mathsf{Mod}})\).

  • Any \({\mathsf{FI}}{\hbox{-}}\)module provides a linear action \(\mathop{\mathrm{End}}_{{\mathsf{FI}}}(n) = S_n\).

• Gradings: functors from \({\mathbb{N}}\to {\mathsf{k}{\hbox{-}}\mathsf{Mod}}\).

Tags: #modular_forms #moduli_spaces #stacks

2021-08-05

Classical / Analytic Moduli Theory

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 \(L_\tau \coloneqq{\mathbb{Z}}+ {\mathbb{Z}}\tau\) 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\coloneqq{\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.

  • One can naturally compactify this by adding the point at infinity to obtain \(X(1) \coloneqq\overline{A_1}\). This point is referred to as a \begin{align*}\[cusp\end{align*} ].

• \(-I\) acts trivially on \({\mathbb{H}}\), so this factors through \(\Gamma \coloneqq{\operatorname{PSL}}_2({\mathbb{Z}}) \coloneqq{\operatorname{SL}}_2({\mathbb{Z}})/\left\langle{\pm I}\right\rangle\).

• 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 \coloneqq{ \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 \(a_{<0} = 0\).

  • \(f\) is a \begin{align*}\[cusp form\end{align*} ] if \(a_0 = 0\).

• 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.

• \(\Gamma(1) \coloneqq{\operatorname{SL}}_2({\mathbb{Z}})\), and principal congruence subgroups of level \(N\) for \(\Gamma(1)\) are defined as \begin{align*} \Gamma(N) \coloneqq\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) &\coloneqq\Gamma\diagdown {\mathbb{H}}^* \\ Y(\Gamma) &\coloneqq\Gamma\diagdown {\mathbb{H}} ,\end{align*} where \({\mathbb{H}}^* = {\mathbb{H}}\cup({\mathbb{Q}}\cup\left\{{\infty}\right\}) \subset {\mathbb{P}}^1({\mathbb{C}})\).

  • Note that \(Y(1)\) parameterizes elliptic curves.

• 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) \coloneqq\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:

  • \(Y_1(N) = Y(\Gamma_1(N))\) is a coarse moduli space for pairs \((E, P)\) where \(P\) is an \(N{\hbox{-}}\)torsion point.
  • \(Y_0(N)\) parameterizes \((E, C)\) where \(C\leq E[N]\) is a cyclic subgroup of the \(N{\hbox{-}}\)torsion points.

• 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.

  • Letting \(\mathrm{Ell} (S)\) be elliptic curves over \(S\) up to isomorphism, \(Y(1)_{/ {{\mathbb{Z}}}} = \operatorname{Spec}{\mathbb{Z}}[j]\) is a coarse moduli scheme for \(\mathrm{Ell} ({-})\), and \(Y(1) = \qty{Y(1)_{/ {{\mathbb{Z}}}} { \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbb{Z}}} }{\times} } \operatorname{Spec}{\mathbb{C}}}^{\mathrm{an}}\) is the associated analytic space.

• 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 \(R \coloneqq{\mathbb{Z}} { \left[ {{1\over 3}, \zeta_3^2} \right] }\) 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 \({\left[ {-1, 0, 1} \right]}, [-1, \zeta_3^2, 0]\).

Moduli as Stacks

• 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.

  • Why elliptic curves have 1-dimensional homology: any globally defined holomorphic 1-form is a double periodic holomorphic 1-form on \({\mathbb{H}}\), forcing it to be constant by Liouville. :

• 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 \(\deg(\sum n_i p_i) \coloneqq\sum n_o\), it turns out that \(\deg((f)) = 0\) for \(f\) principal.

  • As a consequence, meromorphic functions have equal numbers of zeros and poles, and 1-forms that are not identically zero can not have zeros.

• 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 \coloneqq\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

2021-06-22

12:47

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*} ] :

Link to Diagram

• \(M_f\to X\to Y\) yields a LES in homotopy \(\pi_* M_f \to \pi_* X \to \pi_* Y\).
• \(X\to Y\to C_f\) yields \(\pi_*(Y, X) = \pi_{*-1} C_f\)?
  • This should be an easy consequence of the LES in homotopy.

Probability Review

Tags: #probability #review_material #undergraduate

Link to Diagram

• Binomial and Poisson distributions tend to normal distributions?
• Normal distribution rule: 68, 95, 99.7.
• \begin{align*} {\mathbb{E}}[r(x)] \coloneqq\int r(x) f(x) \,dx && \text{or } \sum_k r(k) P(x=k) .\end{align*}
• \({\mathsf{Var}}(x) = E[x^2] - E^2[x]\)
• Marginal densities: if \(f(x, y)\) is a distribution, \begin{align*} f_x(x) \coloneqq\int f(x, y) \,dy && P(x\times y \in A) = \iint_A f(x, y) \,dx\,dy .\end{align*}
  • Independent if \(f(x, y) = f_x(x) f_y(y)\).
• If \(x\) has density \(f_x\) and \(y \coloneqq r(x)\) for a continuous increasing function, setting \(s \coloneqq r^{-1}\), \(y\) has density \begin{align*} g_y(y) = f_x(s(y)) s'(y) .\end{align*}
• Given a failure rate \(\lambda\) in failures over time and a time span \(t\), the probability of failing \(k\) times in \(t\) units of time is \(\operatorname{Poisson}(\lambda t)\).
  • The time between failures is \(\operatorname{Exponential}(\lambda)\).

Combinatorics Review

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 \(i\).

  • Lay out \(n\) copies of 1 with \(n-1\) blanks between them. Choose to put a plus sign or a comma in each slot: choose \(k-1\) commas to produce \(k\) blocks.

• Prop: the number of \begin{align*}\[compositions\end{align*} ] of \(n\) into any number of parts is \(2^{n-1}\).

  • Sum over compositions with exactly \(k\) parts and use the identity \(\sum_{k=1}^{n-1} {n-1 \choose k-1} = 2^{n-1}\).

• Prop: the number of weak compositions of \(n\) into exactly \(k\) parts is \({n+k-1 \choose k-1}\).

  • Add one to each piece to get a (strong) composition of \(n+k\) into \(k\) parts.

• 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.

  • Prop: recurrence for \begin{align*}\[partitions\end{align*} ] : \begin{align*}s(n+1, k) = s(n, k-1) + ks(n, k).\end{align*}
    • Proof: check if a given partition of \(n+1\) into \(k\) parts contains the singleton \(\left\{{n+1}\right\}\).
      • If so, delete and count partitions of \(n\) into \(k-1\) parts.
      • If not, add \(\left\{{n+1}\right\}\) to any of the \(k\) parts.

2021-06-20

21:57

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*} ].

  • How to understand it: look at \begin{align*}\[Galois descent\end{align*} ] on \({\mathsf{K}}({\mathbb{Z}})\)

• 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.

2021-06-06

12:12

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*}

  • Why the first argument: homotopy invariance as a presheaf

• AG setting: \(\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Sch}}_{/k})\), send to presheaves to define \begin{align*}\[motivic spaces\end{align*} ].

  • Satisfies a Nisnevich gluing condition and \({\mathbb{A}}^1\) invariance Similar homotopy invariance: \(F({\mathbb{A}}^1 \times X)\cong F(X)\).

• 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 \({\mathbf{B}}\operatorname{GL}_n \cong {\operatorname{Gr}}_n\).

  • Would like this for non-smooth non-affine \begin{align*}\[schemes\end{align*} ]?

• \begin{align*}\[Algebraic K theory\end{align*} ] : finitely generated projective \(R{\hbox{-}}\)modules mod equivalence with \(\oplus\), then take group completion to get \(K_0(R)\).

  • \(K_0(k) \cong {\mathbb{N}}^ {\operatorname{gp} } \cong {\mathbb{Z}}\).

• To get a space: take \(K(R) \coloneqq\mathop{\mathrm{Proj}}({\mathsf{R}{\hbox{-}}\mathsf{Mod}})^ {\operatorname{gp} }\) to get a space, set \(K_i R \coloneqq\pi_i K(R)\)

• 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.

    • Thick point: point with a tangent direction.

• Burt’s proof worked!

• \begin{align*}\[Grassmannian\end{align*} ] : parameterizes vector bundles with an embedding into \(\infty\)?

  • Cool fact for manifolds: \({\operatorname{Emb}}(M, {\mathbb{R}}^\infty)\) is contractible!
  • \begin{align*}\[stacks\end{align*} ] : presheaves of \begin{align*}\[groupoids\end{align*} ]?

• Some analogs of these theorems for:
  • \begin{align*}\[Hermitian K theory\end{align*} ] :
    • Use orthogonal Grassmannian, take vector bundles with extra data of nondegenerate symmetric \begin{align*}\[bilinear form\end{align*} ]. Need \(\char k \neq 2\). Take \begin{align*}\[Gorenstein\end{align*} ] closed subschemes, which is extra data of orientation.
• \begin{align*}\[Twisted algebraic K theory\end{align*} ] (WIP)
  • Twisted with respect to an \begin{align*}\[Azumaya algebra\end{align*} ] or \begin{align*}\[Brauer class\end{align*} ].

Talbot, Mike Hill

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*}

  • Cohomology of \(H{\mathbb{F}}_2\) is the \begin{align*}\[Steenrod algebra\end{align*} ]?
  • Can compute \(\operatorname{Ext}\), Brunner did this on a computer

• 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!!!

  • Important for Talbot to be a safe space to not necessarily be an expert

• \begin{align*}\[Formal group|formal group laws\end{align*} ] over \(R\): a power series \(x +_F y \coloneqq F(x, y) \in R { \left[ {x, y} \right] }\) such that

  • \(x +_F 0 = x\)
  • \(x +_F y = y +_F x\)
  • \(x+_F (y +_F z) = (x +_F y) +_F z\).

• 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.

  • Having \(\operatorname{ht}\leq n\) is a closed condition, since asking for \(v_{\leq n}\) to vanish is a Zariski closed condition.

• 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 \(F\).

  • Can obtain \(\widehat{G}\) as a \(\colim \operatorname{Spec}k[x] / x^n\), i.e. a formal version of the \begin{align*}\[group scheme\end{align*} ] whose group law is given by \(F\), so if \(F=x+y+xy\) then \(\widehat{G}\) is the \begin{align*}\[formal completion\end{align*} ] of \({\mathbb{G}}_m\) at the identity.

• Examples:

  • \({\mathbb{Z}}/p^n\to {\mathbb{Z}}/p\)
  • \({\mathbb{Z}}[{ {u}_1, {u}_2, \cdots, {u}_{k}} / \left\langle{ p, { {u}_1, {u}_2, \cdots, {u}_{k}} }\right\rangle^m \to {\mathbb{Z}}/p\)

• 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] } \coloneqq 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

  • Even periodicity: \(\pi_{2m+1} E(k, F) = 0\) and \(\pi_{2m+2} E(k, F) \cong \pi_{2m} E(k, F)\)
  • \(\pi_0 E(k, F) = E(k, F)_0\).

• 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 &\coloneqq E({\mathbb{F}}_{p^n}, F_{\mathrm{Honda}}) \\ {\mathbb{G}}_n &\coloneqq{ \mathsf{Gal}} ({\mathbb{F}}_{p^n}) \rtimes S_n \\ S_n &\coloneqq\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 \coloneqq{\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 \({\mathbb{F}_{p^n}}\).

  • It turns out that \(\mathop{\mathrm{Aut}}_(F_{\mathrm{\operatorname{Honda}}}) = {\mathcal{O}}_n^{\times}\)

  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*} ].

  • Use a map \begin{align*} S^0 &\to {\mathsf{E} {\operatorname{O}}}_{p-1}^{hC_p} \\ \beta_{p^i / p^i} &\mapsto \text{non-permanent cycles} .\end{align*}

• Exact same argument for \begin{align*}\[Kervaire invariant 1\end{align*} ] : \(p=2\) version of this argument?

2021-06-05

Talbot, Lyne Moser Part 1

• \({ \underset{(\infty, {2})}{ \mathsf{Cat}} }\): 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*}\[Quasicategory\end{align*} ] : lifting property with lifting against inner horns online

  \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*} ]

  • Latter is more universal, only requires pullbacks and \begin{align*}\[cotensor\end{align*} ] to define?

• 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.

12:58

Things to look up from written notes:

• \begin{align*}\[Picard bundle\end{align*} ]
• \begin{align*}\[Hasse invariant\end{align*} ]
• \begin{align*}\[Connection on a bundle\end{align*} ]
• \begin{align*}\[Frobenius lift\end{align*} ]
• \begin{align*}\[lambda ring\end{align*} ]
• \begin{align*}\[absolute Galois group\end{align*} ]
• For \begin{align*}\[elliptic curve|elliptic curves\end{align*} ] :
  • \begin{align*}\[level of an elliptic curve\end{align*} ], \begin{align*}\[weight of an elliptic curve\end{align*} ], \begin{align*}\[conductor of an elliptic curve\end{align*} ]
• What is the difference between \begin{align*}\[local class field theory\end{align*} ] and \begin{align*}\[global class field theory\end{align*} ]
• \begin{align*}\[theta function\end{align*} ]

Talbot, Lyne Moser Part 2

• Next: models of \({ \underset{\infty}{ \mathsf{Cat}} }{\infty, 2}\)

  • Can take \begin{align*}\[enrichment\end{align*} ] over \({ \mathsf{quasiCat} }\) or \({ \mathsf{CSS} }\)

• \begin{align*}\[base change along a functor\end{align*} ]??

• \begin{align*}\[2-category\end{align*} ] : categories \begin{align*}\[enriched\end{align*} ] in categories

• Recall: \({ \mathsf{CSS} } = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{sSet}})\).

  • We have \(\Delta \subseteq \mathsf{Cat}\) a full subcategory, we now want a version for \(2{\hbox{-}}\mathsf{Cat}\): \(\Theta_2^{\operatorname{op}}\).
  • Turns out to be \begin{align*}\[wreath product\end{align*} ] \(\Delta\wr\Delta\).

• 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*} ]

2021-06-04

Random Physics Reading?

• 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 \(Y(A, z) = e^{zT} A = \sum _{T^k A \over k!} z^k\).

  • Then \(A_{(n)}\) is given by multiplication in \(V\) of the form \({1\over (n-1)!} T^{-n-1} A\).

• 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?

  • Special case: for \(V\) a rational vertex algebra, its \begin{align*}\[representation category\end{align*} ] is modular tensor.

• Beilinson-Drinfeld, 90s: factorization/chiral algebras

  • \begin{align*}\[Factorization spaces\end{align*} ] : an assignment of spaces \(\mathcal{Y}_n \to X^{\times n}\) for \(X\), Ran’s condition on the inclusion \(\Delta X\to X^{\times 2}\), and factorization isomorphisms, conditions on \(\diagonal^c\).
  • For factorization algebras, make the assignment a \begin{align*}\[sheaf\end{align*} ].

• 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*} ].

  • For a smooth curve \(X\) and \(G\) a \begin{align*}\[reductive group\end{align*} ], built out of \begin{align*}\[principal bundle|principal G-bundles\end{align*} ].
  • Parameterizes triples \(\mathbf{x}\in X^n\), \(\sigma\) a principal \(G\) bundle, and \(\xi\) a trivialization of \(\sigma\) in \(X\setminus\mathbf{x}\).
  • Important in \begin{align*}\[geometric Langlands\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).

  • Can get a sheaf out of this which is a chiral algebra over the curve.

• 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!

2021-05-25

12:03

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*} ]

  • There is a sensible way to define Brauer groups for formal schemes as a holim

• \(\lim^1\), 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 \(A\otimes A' \sim R\)

  • Can identify \({\mathsf{Fun}}(\mathsf{A}{\hbox{-}}\mathsf{Mod}, \mathsf{B}{\hbox{-}}\mathsf{Mod}) \cong ({A^^{\operatorname{op}}}, {B}){\hbox{-}}\mathsf{biMod}\)

• 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*} ]

  • See \begin{align*}\[formal GAGA\end{align*} ] for perfect complexes

2021-05-24

12:07

Reference: ???, GROOT

• Big idea: free implies flat for algebras, is this true in the equivariant settings?

  • Almost all something are something, check talk title!!

• 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*} ].

  • Composition of spans is pullback.
  • Sends disjoint unions

    \to

    direct sums
  • Every object is the disjoint union of orbits \(G/H\)

• 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.

  • Are transfers like inflation? #unanswered_questions

• \begin{align*}\[Burnside Mackey functor\end{align*} ] : \(\underline{A}\).

  • Objects are \(K_0\) of finite groups under \(\coprod\), \(\operatorname{res}\) is the forgetful functor, \({\mathrm{tr}}_K^H([x]) = [H { \underset{\scriptscriptstyle {K} }{\times} } X]\).

• 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.

  • A Mackey functor \(R\) where \(R(G/H)\) is a unital commutative ring and \(\operatorname{res}^H_K\) is a ring morphism.

• 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.

  • \(\underline{A}\) has norms given by \({\mathsf{Set}}^K(A, B)\), \(K{\hbox{-}}\)equivariant set functions.

• 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.

17:45

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?

  • I’ve been thinking about “integrating over a category” a lot, some way to extract “average information” about a category. Integration on moduli spaces is hard!

• Look up \begin{align*}\[simple normal crossings\end{align*} ] divisor.

21:36

• 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*} ]?

2021-05-22

23:27

Tags: #idle_thoughts #category_theory #infinity_cats

Cats with cats of morphisms

Terrible attempt at a way around ZFC: axiomatically define a category the way one axiomatizes Euclidean geometry:

• A collection of points (objects)
• For every two points, a category of morphisms
  • Plus the usual composition axiom

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*} ].

Set as a category freely generated under colimits?

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?

2021-05-18

Kristen Hendricks, Surgery formulas for involutive Heegaard Floer homology

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*} ] \(\Theta_{\mathbb{Z}}^3\).

  • We don’t understand the torsion in this group.
  • Reduce to study of a group of “\(\iota\) complexes”.
  • Theorem: it has a \({\mathbb{Z}}^{\infty}\) summand.

• 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*} ].

22:01

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*} ] :

  attachments/image_2021-05-18-22-20-05.png❗
  • Interpretation of the triangle: \(F = E \ominus {\mathcal{O}}(-n) = E \oplus {\mathcal{O}}(-n)[1]\).
  • Interpret \({\mathcal{O}}(-n)[1] = -{\mathcal{O}}(n)\).

• What is an \begin{align*}\[etale algebra\end{align*} ]?

2021-05-13

00:14

• \(\pi_1(X)\) can be defined for schemes, see \begin{align*}\[etale fundamental group\end{align*} ].
• What are the higher homotopy groups of schemes? #unanswered_questions
  • What do they measure?
  • More fundamentally, what do higher homotopy groups of spheres measure about \({\mathsf{Top}}\) at all??
• What are the \begin{align*}\[Eilenberg-MacLane spaces\end{align*} ] in \({\mathsf{Sch}}\)?
• What are the Moore spaces in \({\mathsf{Sch}}\)?
  • There should be an axiomatic characterization of these coming from \begin{align*}\[Model category theory|model categories\end{align*} ]
• What are the “spheres” for \({\mathsf{Sch}}\)?

2021-05-12

10:28

See \begin{align*}\[Serre's uniformity conjecture\end{align*} ].

Pavel Etingof, Frobenius exact symmetric tensor categories

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.

  • \begin{align*}\[irreducible\end{align*} ] representations: hard, but a lot is known.
  • \begin{align*}\[indecomposable objects of a category\end{align*} ] representations: very hard, very little is known. Hard

• 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*} ] \(\mathsf{C}\):

  • \(k{\hbox{-}}\)linear, so enriched in \({ \mathsf{Vect} }_{/k}\) for \(k= \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) (no assumption on characteristic) So morphisms are vector spaces and composition is bilinear.
  • \begin{align*}\[abelian category\end{align*} ]
  • \begin{align*}\[Artinian category\end{align*} ] : objects have finite length and \(\dim_K \mathsf{C}(X, Y) < \infty\)
  • \begin{align*}\[Monoidal category\end{align*} ] : \((\otimes, \one)\) satisfying associativity and the pentagon axiom
  • \begin{align*}\[Symmetric monoidal category\end{align*} ] : a symmetric braiding \(X\otimes Y \xrightarrow{\tau_{XY}} Y\otimes X\) such that \(\tau_{YX} \circ \tau_{XY} = \operatorname{id}\).
  • \begin{align*}\[rigid category\end{align*} ] : existence of duals and morphisms \(X\to X {}^{ \vee }\) plus rigidity axioms
  • Compatibility of additive/multiplicative structures: implied when \(\otimes\) is bilinear on morphisms.
  • \(\mathop{\mathrm{End}}_{\mathsf{C}}(\one) = k\).

• Example: \({\mathsf{Rep}}_k(G)\) the category of finite-dimensional representations of \(G\). Take \(G=1\) to recover \({ \mathsf{Vect} }_{/k}\).

  • Can replace a group here by an affine \begin{align*}\[group scheme\end{align*} ]

• Such a category is \begin{align*}\[tannakian\end{align*} ] if there exists a \begin{align*}\[fiber functor\end{align*} ] : a symmetric tensor functor \(F: \mathsf{C} \to { \mathsf{Vect} }_{/k}\).

  • Preserves tensor structure (hexagon axiom), preserves braiding, is exact.
  • Implies automatically faithful.
  • Can take forgetful functor from representations to underlying vector space.
  • Called “fiber” because for spaces and local systems, one can take a fiber at a point
  • Deligne-Milne show this is unique.
  • Can define scheme of tensor automorphisms, \(G = \underline{\mathop{\mathrm{Aut}}}^\otimes(F) \in {\mathsf{Grp}}{\mathsf{Sch}}_{/k}\).

• 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*} ] \(\operatorname{Tr}(f) \in \mathop{\mathrm{End}}_{\mathsf{C}}(\one)\):

Link to Diagram

• Can define a categorical \begin{align*}\[dimension of a category\end{align*} ] \(\dim X \in K\) as \(\dim X\coloneqq\operatorname{Tr}(\operatorname{id}_X)\).
• See theory of \begin{align*}\[semisimplification\end{align*} ].
• A morphism \(f\in \mathsf{C}(X, Y)\) is negligible if for all \(g\in \mathsf{C}(Y, X)\) we have \(\operatorname{Tr}(f\circ g) = 0\).
  • Negligible morphisms form a \begin{align*}\[tensor ideal\end{align*} ] of morphisms: stable under composition and tensor product with other morphisms.
• Can form quotient \(\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu \mathsf{C} / \mathsf{N}\): full subcategory where \(\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu(X, Y) \coloneqq\mathsf{C}(X, Y) / \mathsf{N}(X, y)\), i.e. form the vector space quotient of the hom sets.
  • Still monoidal.
  • Generally nasty, but if trace of any nilpotent endomorphism in \(\mathsf{C}\) (e.g. when \(\mathsf{C}\) admits a monoidal functor to an abelian STC), then \(\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu\) is a \begin{align*}\[semisimple category\end{align*} ] STC, and in particular is abelian and every object is a direct sum of simple objects.
  • True if \(\mathsf{C}\) is abelian: take any nilpotent endomorphism, filter by kernels of powers and take the associated graded. Trace of original equals trace of associated graded, but the latter is zero.
  • Can compute trace after pushing through an abelian functor.
  • \(\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu\) is the \begin{align*}\[semisimplification\end{align*} ] of \(\mathsf{C}\)
• \begin{align*}\[simple objects of a category\end{align*} ] in \(\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu\) are \begin{align*}\[indecomposable objects of a category\end{align*} ] in \(\mathsf{C}\) of nonzero dimension.
  • This procedure is forcing \begin{align*}\[Schur's Lemma\end{align*} ] to be true!
• Run into functors that aren’t exact on either side, e.g. the Frobenius.
  • But still additive, and every additive functor on a semisimple category is exact.
• Frobenius functors: take \(X^{\otimes p}\), allow cyclic permutations \(c\) with \(c^p = 1\), get \begin{align*}\[equivariant|equivariance\end{align*} ] with respect to \({\mathbb{Z}}/p\).
  • Thus \(X^{\otimes p} \in \mathsf{C} \boxprod {\mathsf{Rep}}_{/k} {\mathbb{Z}}/p\), the \begin{align*}\[Deligne tensor product\end{align*} ].
  • Can take \(\operatorname{id}_{\mathsf{C}} \boxprod \ss\), i.e.  \begin{align*}\[semisimplification\end{align*} ] on the 2nd component, to get an additive monoidal twisted-linear functor \(\mathop{\mathrm{Fr}}: \mathsf{C} \to \mathsf{C} \boxprod \Ver_p\)
    • Here \(\Ver_p\) is a Verlinde category: just denotes \(\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu\) with a modified tensor product formula…?
• Can \begin{align*}\[filtration\end{align*} ] \(X^{\otimes p}\) by \(\mathop{\mathrm{Fr}}_i \coloneqq\ker (1-c)^i\).

2021-05-11

12:48

• 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?

  • How is this related to \begin{align*}\[arithmetic fracture squares\end{align*} ]?
  • What about passing from \begin{align*}\[adelic\end{align*} ] solutions instead?

• 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*} ].

2021-05-10

12:34

We haven’t been able to classify the rational points on \begin{align*}\[modular curves\end{align*} ]!

Kirsten Wickelgren, Zeta functions and a quadratic enrichment.

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*}\[dualizable|dualizable objects in a category\end{align*} ]

Link to Diagram

• Works more generally for a \begin{align*}\[symmetric monoidal category\end{align*} ]
• Finite dimensionality is replaced by objects being \begin{align*}\[dualizable\end{align*} ], so for \begin{align*} \one & \xrightarrow{m} A\otimes B \\ B\otimes A & \xrightarrow{{\varepsilon}} \one ,\end{align*} require

Link to Diagram

• 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*} ] :

Link to Diagram

• 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 \(X \coloneqq{\mathbb{P}}^n/{\mathbb{P}}^{n-1}\), 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*}

  • Recovers degree on \(X({\mathbb{C}})\).

• \begin{align*}\[Grothendieck-Witt\end{align*} ] group: formal differences of isomorphism classes of nondegenerate symmetric \begin{align*}\[bilinear forms\end{align*} ].

  • Allow orthogonal direct sum and orthogonal direct difference.

• 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*} ] :

  • Define \({\mathsf{K}}_0({\mathsf{Var}}_k)\) to be the group completion of varieties under cut-and-paste
  • Define \begin{align*} Z_X^m(t) \coloneqq\sum_{m\geq 0} [\operatorname{Sym}^m X] t^m \in {\mathsf{K}}_0 ({\mathsf{Var}}_k) {\left[\left[ t \right]\right] } .\end{align*}
  • Define an \begin{align*}\[Euler characteristic\end{align*} ] \begin{align*} \chi_C^{{\mathbb{A}}^1}: K_0({\mathsf{Var}}_k) \to {\operatorname{GW}}(k) .\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:

• Defines a \begin{align*}\[zeta function\end{align*} ] for any endomorphism of any variety. Doesn’t need to be over \({\mathbb{F}}_p\), and doesn’t need to have Frobenius!

Foling Zou, Nonabelian Poincare duality theorem and equivariant factorization homology of Thom spectra

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.

• \begin{align*}\[factorization homology\end{align*} ] setup:

• 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.

  • Another example: the \begin{align*}\[Kelly product\end{align*} ] :

• Can define operads and reduced \begin{align*}\[operads\end{align*} ] as \begin{align*}\[monoids\end{align*} ] in certain categories:

• 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.

  • Also appears as a step in a later proof identifying \({\operatorname{THH}}_{C_2} ({ \mathsf{H} }{\mathbb{F}}_2) \approx { \mathsf{H} }{\mathbb{F}}_2 \wedge({\Omega}S^3)_+\).
  attachments/image_2021-05-10-17-27-00.png❗

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*} ]?

2021-05-08

22:55

Not every \begin{align*}\[simplicial complex\end{align*} ] is a \begin{align*}\[PL\end{align*} ] manifold:

2021-05-06

\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*} ]

2021-05-05

01:42

• There is apparently a theory of \begin{align*}\[algebraic cobordism\end{align*} ].

Padmavathi Srinivasan, UGA NT Seminar

^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 \(p\),

  • Want to find rational points
  • If \(r<g\) (?) then \begin{align*}\[Chabauty-Coleman\end{align*} ] applies.
  • This talk: \(r=g\), allows finding a basis for \({\mathbb{Q}}_p\) valued functions on \(J({\mathbb{Q}})\).

• See p-adic \begin{align*}\[height pairing\end{align*} ].

  • Height pairing has to do with rational points on \begin{align*}\[Jacobian\end{align*} ]?
  • Trying to compute \begin{align*}\[canonical heights\end{align*} ]?
  • See \begin{align*}\[global height\end{align*} ] vs \begin{align*}\[local height\end{align*} ]
  • See \begin{align*}\[Weil height machine\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.

  • Need a curvature form in \(\Omega^1(X) \otimes H^1_\mathrm{dR}(X)\).
  • Get one height for each choice of \begin{align*}\[idele class character\end{align*} ]

• 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.

  • It’s a locally bounded continuous function \(\nu: \mathcal{L}^{\times}\coloneqq{ \operatorname{Tot} }( \mathcal{L} ) \setminus\left\{{0}\right\}\to {\mathbb{R}}\) with \(\nu( \alpha \mathbf{v}) = v_p(\alpha) + \nu(\mathbf{v})\) for \(\alpha\in \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{Q}}_p\mkern-1.5mu}\mkern 1.5mu^{\times}, \mathbf{v} \in \mathcal{L}^{\times}\).
  • Here continuous is in the locally analytic topology, since we’re over \({\mathbb{Q}}_p\).
  • Can do all of the usual stuff carrying the additional data of the metric: tensor powers, pullbacks, etc.

• When you have an integral model: take closures!

  • Picking integral models allows measuring sizes of sections.

• 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*} ]

  • See \begin{align*}\[semistable model\end{align*} ] of a curve \(X_{/\QQp}\).

• Adelic metric: a collection of metrics for almost every \begin{align*}\[place\end{align*} ].

  • Associated height function: for \(x\in X({\mathbb{Q}})\), pick a section not vanishing at \(x\) and sum all contributions: \(x\mapsto \sum_{p \in {\operatorname{Places}}} v_p (s(x))\).

• 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*} ]

  • Apply Banach fixed-point theorem to the self-map \(A \xrightarrow{[2]} A\).

• 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 \(\omega\).

2021-05-04

The reciprocity law for the twisted second moment of Dirichlet L-functions

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*} ]?

The K-Theory of monoid sets

Reference: The \(K'\)-theory of \begin{align*}\[monoid\end{align*} ] sets https://arxiv.org/pdf/1909.00297.pdf

\begin{align*}\[K-Theory\end{align*} ]

• \(K'(A)\) defined for partially cancellative \(A{\hbox{-}}\)sets.
  • Important example: the pointed \begin{align*}\[monoid\end{align*} ] \({\mathbb{N}}\coloneqq\left\{{{\operatorname{pt}}, 1, t, t^2, \cdots, }\right\}\).
• Useful in \begin{align*}\[toric geometry\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 \(K({\mathsf{FinSet}}_{\scriptstyle { * } }) \simeq{\mathbb{S}}\).

  • If \(A\) has no idempotents or units then \(K(A) \simeq{\mathbb{S}}\).

• \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 \({\mathsf{K}}(A) \cong {\mathsf{K}}(B)\)?

• Cancel all of the things:

• Apparently easy theorem: \({\mathsf{K}}'({\mathbb{N}}) \simeq{\mathbb{S}}\).

• The \begin{align*}\[Picard group\end{align*} ] of \({\mathbb{P}}^1\) shows up:

Stefan Schreieder, Refined unramified cohomology

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 \(Z^i(X)\)?

  • See algebraic equivalence in the \begin{align*}\[Chow group\end{align*} ].

• \begin{align*}\[Gysin sequence\end{align*} ] yields a residue map \({\partial}_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)\).

• See \begin{align*}\[Gersten conjecture\end{align*} ]

• 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.

  • \begin{align*}\[pro-scheme\end{align*} ] : an inverse limit of \begin{align*}\[scheme\end{align*} ].

• 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.

  • Use an \begin{align*}\[Enriques surface\end{align*} ] to produce \(({\mathbb{Z}}/2)^{\oplus \infty}\) in \(\mathop{\mathrm{Griff}}^3\).

• 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.

  • Dual \(\beta\) of \([K_S] \in H^2(S; {\mathbb{Z}}/2)\) generates the \begin{align*}\[Brauer group\end{align*} ] \(\mathop{\mathrm{Br}}(S)\) of the surface. Note \(\beta\) is not algebraic.

• 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 \(\mathop{\mathrm{Br}}(S) \twoheadrightarrow\mathop{\mathrm{Br}}(S_\eta)\).

  • \(\mathop{\mathrm{Br}}(X_\eta) \cong {\mathbb{Z}}/2\) is generated by an \begin{align*}\[unramified\end{align*} ] conic bundle.
  • Can extend conic smoothly over \begin{align*}\[central fiber\end{align*} ]
  • Need that the Poincaré dual \begin{align*}\[specializes\end{align*} ] to zero on the \begin{align*}\[special fiber\end{align*} ].

• 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*} ].

Clausen on rep theory

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*} ].

  • Need the \begin{align*}\[equivariant derived category\end{align*} ], very difficult to define!

2021-05-03

Representations of Hopf Algebras

Tags: #representation_theory

See \begin{align*}\[Hopf algebra\end{align*} ]

• Algebras: \(m: A^{\otimes 2} \to A\) and \(u:k\to A\) the unit with associativity:

Link to Diagram

• \begin{align*}\[Coalgebras\end{align*} ] : \(\Delta: A\to A^{\otimes 2}\), \({\varepsilon}: A\to k\) the counit. Reverse the arrows in the diagram for coassociativity. This yields a bialgebra, for Hopf structure need an antipode \(s:M\to M\):

Link to Diagram

Why \begin{align*}\[Hopf algebra\end{align*} ]? Some natural examples:

• \(kG\) the \begin{align*}\[group algebra\end{align*} ].

  • \(\Delta(g) \coloneqq g^{\otimes 2}\)
  • \({\varepsilon}(g) = 1_G\)
  • \(s(g) = g^{-1}\)

• \(k^G = \mathop{\mathrm{Hom}}_k(kG, k)\) 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 \(H\).

  • Issue: tensor product of \(R{\hbox{-}}\)modules may not again be an \(R{\hbox{-}}\)module.

• 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*} ].

  • Evaluation \(M {}^{ \vee }\otimes M \to \one\) and coevaulation \(\one \to X\otimes X {}^{ \vee }\).
    • For finite dimensional vector spaces, \(k\mapsto \sum k e_i \otimes e_i {}^{ \vee }\)?
  • Finite rank: finitely many simples up to isomorphism. Can still have infinitely many indecomposables.

• 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 \(H^*(\mathsf{C}) \coloneqq H^*_{\mathsf{C}}(\one, \one) = \bigoplus _{n\geq 0} \operatorname{Ext} ^n_{\mathsf{C}} (\one, \one )\). 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*} ].

Clausen, the K-theory of adic spaces.

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 \(R = \varprojlim R/I^n\)

  • Yields \begin{align*}\[scheme\end{align*} ] as a subcategory?

• Some nice features:

  • Topological bases of \begin{align*}\[quasicompact\end{align*} ] open subsets
  • Has a nice ring attached to each subspace.

• Subtleties:

  • Structure sheaf is only a presheaf and not necessarily a sheaf
  • Not even great when it is a sheaf: can’t work locally

• \(\mathsf{Solid}_{\mathbb{Z}}\): 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 \(\prod_I {\mathbb{Z}}\)

  • \begin{align*}\[Compact generators\end{align*} ] : mapping out to filtered colimits..?
  • \begin{align*}\[Projective generators\end{align*} ] : lift along surjections
  • \begin{align*}\[Generators of a category\end{align*} ] : everything is a cokernel of direct sums of these

• 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 \(K(\mathsf{Perf}(X))\)?

  • Doesn’t satisfy \begin{align*}\[descent\end{align*} ]

• 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*} ]?

  • Something to do with \begin{align*}\[arithmetic surfaces\end{align*} ].
  • Some apparent contributions by Faltings:
    • A \begin{align*}\[Riemann-Roch\end{align*} ] theorem
    • A \begin{align*}\[Noether formula\end{align*} ]
    • A \begin{align*}\[Hodge index theorem\end{align*} ]
    • Non-negativity of the self-intersection of the \begin{align*}\[dualizing sheaf\end{align*} ].
  • Vojta 1991: new proof of the \begin{align*}\[Mordell conjecture\end{align*} ]

2021-05-02

Hopf Invariant One

• Cool fact: you can show the non-existence of \begin{align*}[Hopf invariant one](#hopf-invariant-one)\end{align*} elements in dimensions \(1,2,4,8\) using \begin{align*}\[K-Theory\end{align*} ].
• The argument is pretty clean, and the paper is only 8 pages:
  • https://www.maths.ed.ac.uk/~v1ranick/papers/adamatiy.pdf
• See Mosh-Tang for the original proof using \begin{align*}\[cohomology operations\end{align*} ] : https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf.
  • This also apparently includes computations of cohomology of \begin{align*}\[Eilenberg-MacLane spaces\end{align*} ].

2021-05-01

Notes on `

• 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

  • For \(G={\mathbb{Z}}/2\), it is \(O(N)\).
  • For \(G={\mathbb{Z}}/3\), it is \(O(\sqrt N )\).

• 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*} ] \(C_{/k}\).

  • The (reverse) functor is the one sending a curve \(C\) to its function field \(k(C)\).

• \begin{align*}\[Hurwitz spaces\end{align*} ] come up here!

• \begin{align*}\[etale cohomology\end{align*} ] :

• \({\varepsilon}\) is a \(q^i\) \begin{align*}\[Weil number\end{align*} ] if \({\left\lvert { \iota({\varepsilon}) } \right\rvert} = q^{i/2}\) for any embedding \(\iota: { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\hookrightarrow{\mathbb{C}}\).

  • Examples: eigenvalues of geometric \begin{align*}\[Frobenius\end{align*} ] acting on \(H^i_c\).

• 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*} ]

• \begin{align*}\[Homological Stability Course Notes\end{align*} ] for \begin{align*}\[Hurwitz spaces\end{align*} ] :

Old Notes: Erik Schreyer

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

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 \(p\).

A SES isomorphic to a direct sum that does not split

Reference

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.

Cogroups

See \begin{align*}\[cogroup\end{align*} ].

2021-04-30

Remy van Dobben de Bruyn, “Constructing varieties with prescribed Hodge numbers modulo m in positive characteristic”.

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 \(n\),

  • Linear relations are spanned by \begin{align*}\[Serre duality\end{align*} ] in positive characteristic.
  • In \(\operatorname{ch}(k) > 0\), the only polynomial relations are \(h^{0,0} = 1\) and Serre Duality.
  • In \(\operatorname{ch}(k) = 0\), one has to add in Hodge symmetry.

• Important tools:

  • Kunneth formula for Hodge diamonds: there’s a graphical way to do this by summing over several different ways to place blocks in the diamonds.
  • See \begin{align*}\[blowup|Blowups\end{align*} ],

\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*} ].

l-adic Representations

• 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*} ]

2021-04-26

Random Notes

Some random notes: #todo

-Working out relative homology, an example: attachments/image_2021-04-25-01-52-05.png❗

• Chain of implications for module properties: attachments/image_2021-04-25-01-52-56.png❗

• Definitions of common matrix groups: attachments/image_2021-04-25-01-53-18.png❗

• Good example of exact triangles: attachments/image_2021-04-25-01-53-49.png❗

• Manifolds from the sheaf perspective, a reference: attachments/image_2021-04-25-01-54-21.png❗

Random Algebraic Topology

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*} ]? #unanswered_questions 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? #unanswered_questions

• 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*} ]. #unanswered_questions

• What is the \begin{align*}\[Stiefel manifold\end{align*} ]? #unanswered_questions

  • I should write down an explicit set-theoretic description somewhere. This is definitely in Fomenko.

19:38

• Is there a natural exact sequence associated to a \begin{align*}\[composition series\end{align*} ]? #unanswered_questions
  • This seems like it should be super easy, we have quotients everywhere.
  • Is there a precise relation to \begin{align*}\[iterated extensions\end{align*} ]..? #unanswered_questions

2021-04-25

Fukaya Category

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 \(\hom_{\mathcal{O}}\coloneqq\qty{\bigoplus_{L_0 \pitchfork L_1} {\mathbb{Z}}/2{\mathbb{Z}}, {{\partial}}}\) where \({{\partial}}\) is given by counting holomorphic strips, localize along small isotopies.

Notes from Eisenbud

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.

Random

• Steenbrink spectral sequence (Peters-Steenbrink for exposition)
• Rapoport-Zink spectral sequence
• Bounding ranks of curves over a \begin{align*}\[Function field\end{align*} ] : see \begin{align*}\[elliptic fibrations\end{align*} ]
• \begin{align*}\[Burnside ring (algebraic geometry)\end{align*} ] in AG: Take the free abelian group on finitely generated field extensions over a base field.
• Check statement of the Baez-Dolan \begin{align*}\[cobordism hypothesis\end{align*} ]

Milnor K Theory in the Wild

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: attachments/image_2021-04-25-12-25-47.png❗

Modular forms and Deligne-Serre theorem

• \begin{align*}\[modular form\end{align*} ] yield 2-dimensional \begin{align*}\[Galois representations\end{align*} ], and there is a classification theorem:

  Deligne-Serre Theorem: attachments/image_2021-04-25-12-43-06.png❗

The representation ring

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*} ].

  • How can we construct this using modern \begin{align*}\[groupoid\end{align*} ] yoga? Take the category \(\mathsf{G}{\hbox{-}}\mathsf{Mod}\), somehow restrict to just \begin{align*}\[irreducibles\end{align*} ]. Maybe there’s a better thing to do here though, like “ignoring” reducibles the same way John Carlson “ignored” projectives. But okay, anyway, take that category. Take its \begin{align*}\[nerve\end{align*} ] and then the \begin{align*}\[geometric realization\end{align*} ] and then \(\pi_0\) or something? And then take the free \({\mathbb{Z}}{\hbox{-}}\)module. I definitely need to ask some homotopy theorists how this construction goes for usual \begin{align*}\[K-Theory\end{align*} ] in modern terms. So like… \begin{align*} {\mathbb{Z}}\left[ \pi_0 {\left\lvert { N \mathsf{C} } \right\rvert} \right] .\end{align*} The \(\pi_0\) should be taking isomorphism classes somehow, but maybe this only works for groupoids? But okay, whatever, I just need a functor that takes categories into spaces where two objects end up in the same path component iff they’re isomorphic in \(\mathsf{C}\). So maybe this needs to be something more \begin{align*}\[simplicial set\end{align*} ].

2021-04-23

What is `

https://arxiv.org/pdf/1305.4293.pdf

• 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*} ].

    • Gauss-Bonnet and Hopf index theorem: attachments/image-20210218021511916.png❗

• What is the \begin{align*}\[Homotopy quotient\end{align*} ]? #unanswered_questions

  • attachments/image-20210218013730610.png❗

• If \(G\curvearrowright M\) with \(G\) a compact connected \begin{align*}\[Lie group\end{align*} ], Cartan constructs a chain complex from \(M, {\mathfrak{g}}\).

  • attachments/image-20210218013828220.png❗

  • Is this not precisely the \begin{align*}\[Borel construction\end{align*} ]? #unanswered_questions

• \begin{align*}\[classifying spaces\end{align*} ] : \({\mathbf{B}}S^1 = {\mathbb{CP}}^{\infty}\)

• Why are \begin{align*}\[maximal torii\end{align*} ] useful?
  • attachments/image-20210218014526989.png❗ #unanswered_questions

What is a scheme?

• 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 \(V(S) \coloneqq\left\{{ q\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}q\supseteq S}\right\}\)

Notes on `

• http://mathieu.anel.free.fr/mat/doc/Anel-Semiomaths-HomotopyColimit.pdf

• attachments/image-20210218010308802.png❗
• attachments/image-20210218010627345.png❗
• attachments/image-20210218010639958.png❗
• attachments/image-20210218010749422.png❗
• attachments/image-20210218010809139.png❗
• attachments/image-20210218011118003.png❗
• attachments/image-20210218012454677.png❗

• Hocolims are \begin{align*}\[infinity groupoids\end{align*} ], equivalently \begin{align*}\[homotopy type\end{align*} ].

• There is a functor \(\pi_0: { \underset{\infty}{ \mathsf{Grpd}} }\to {\mathsf{Set}}\).

15:07

• Hironaka: Fields for existence of \begin{align*}\[Resolution of singularities\end{align*} ] in every dimension in \(\operatorname{ch}(k) = 0\).

\begin{align*}\[2021-04-23 Advice on research and problems\end{align*} ]

Time Management

• 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.

SeZoom

\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.

  • Easy for \begin{align*}\[elliptic curve\end{align*} ].
  • For \begin{align*}\[moduli stack of abelian varieties\end{align*} ], requires \begin{align*}\[Néron models\end{align*} ], see Silverman.

• \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.

17:13

• Relation between \begin{align*}\[quadratic form\end{align*} ] and unique factorization:

22:29

• See Marcus (?) for a nice proof of \begin{align*}\[quadratic reciprocity\end{align*} ] involving looking at primes splitting in \begin{align*}\[quadratic fields\end{align*} ].

2021-04-22

Gromov-Witten Invariants in Derived AG

• My main question: does introducing derived stacks somehow make some computation easier? #unanswered_questions

  • I haven’t found any explicit computations of these, but sources alluded to actual counts (numbers) conjecturally coming from physics, where a few have been verified.

• Integrating over a \begin{align*}\[fundamental class\end{align*} ] :

• \begin{align*}\[moduli space\end{align*} ] of \begin{align*}\[stable maps\end{align*} ]

• \begin{align*}\[Operads\end{align*} ] :

• Appearance of \begin{align*}\[Calabi-Yau\end{align*} ] in \begin{align*}\[Physics\end{align*} ]

  • Related: \begin{align*}\[String theory\end{align*} ], \begin{align*}\[Mirror Symmetry\end{align*} ]
  attachments/image_2021-04-22-12-12-17.png❗

• Mirror symmetry of CYs:

  attachments/image_2021-04-22-12-12-51.png❗

• The major types of “moduli” style invariants

  attachments/image_2021-04-22-12-13-46.png❗
  • See \begin{align*}\[quantum invariants\end{align*} ]

• Why care about \begin{align*}\[coherent sheaves\end{align*} ]? #unanswered_questions

• \begin{align*}\[Donaldson-Thomas invariants\end{align*} ] are supposed to relate to \begin{align*}\[Gromov-Witten invariants\end{align*} ] :

  attachments/image_2021-04-22-12-17-02.png❗

• Niceness of spaces:

  attachments/image_2021-04-22-12-17-44.png❗

Derived Stacks

• 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.

    • Morphisms behave like they are transverse?

• 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 \(X\).

  • The homology of its “tangent space” at a point \([E]\) is \(\operatorname{Ext} ^*(E, E)\), which only holds in restricted degrees if you only use a non-derived moduli scheme or stack.

2021-04-21

15:05: Bhargav Bhatt (Harvard NT Seminar)

#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.

  • \(I {~\trianglelefteq~}A\) is an ideal defining a \begin{align*}\[Cartier divisor\end{align*} ].
  • \(A\) is \((P, I){\hbox{-}}\)complete.
  • Any ideal generator \(d\in I\) satisfies \(\phi(d) = d^p + p\cdot u, u\in A^{\times}\).

• Fix a scheme and study prisms over it. Need these definitions to have stability under base-change.

• Examples:

  • \(A \coloneqq{\mathbb{Z}}_p\) and \(\phi = \operatorname{id}\) with \(I = \left\langle{ p }\right\rangle\) yields \begin{align*}\[Crystalline cohomology\end{align*} ].
  • \(A \coloneqq{\mathbb{Z}}_p{\left[\left[ u \right]\right] }, \phi(u) = u^p\). Then \(I = \left\langle{ E(u) }\right\rangle\) is generated by an \begin{align*}\[Eisenstein polynomial\end{align*} ]. Here \(A/I = {\mathcal{O}}_K\)

• 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 \coloneqq\left\{{ (B, J) \in \mathsf{Prism},\, \operatorname{Spf}(B/J) \to X }\right\} .\end{align*} Take \begin{align*}\[sheaf cohomology\end{align*} ] to obtain \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism(X) \coloneqq\mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_\prism, {\mathcal{O}}_\prism) {\circlearrowleft}_\phi\).

• 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.

  • What is the \begin{align*}\[cohomological dimension\end{align*} ] of \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism(X)\)?
  • What are the \(F{\hbox{-}}\) \begin{align*}\[crystals\end{align*} ] on \(X_\prism\)?
    • Produce finite flat \(B{\hbox{-}}\)modules?

• 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*}

  • Translates other cohomology theories into something about coherent sheaves..?
  • Can reduce to studying e.g. a vector bundle on a more complicated object.

• Def: Cartier-Witt stack, a.k.a. the prismatization of \({\mathbb{Z}}_p\)

  • Define \(\mathsf{WCart}\) to be the \begin{align*}\[formal stack\end{align*} ] on \(p{\hbox{-}}\)complete rings.

• 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*} ].

  • Receives a natural Frobenius action, which is a \begin{align*}\[derived Frobenius lift\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 \((W(k), ?)\).

  • Yields a (geometric?) point \(x_{{\mathbb{F}_p}}: \operatorname{Spec}({\mathbb{F}_p}) \to \mathsf{WCart}\).

• 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

Link to 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*}

  • Fact: \(\pi_{\operatorname{HT}}\) is a flat cover and \(\mathop{\mathrm{Aut}}(\pi_{\operatorname{HT}}) = W^*[ F]\).

• 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 \({\mathsf{gr}\,}_i = \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X, \Omega^i_X)[-i]\).

  • Use representation interpretation, then \(\mu_p \curvearrowright{\mathsf{gr}\,}_i\) by weight \(-i\).

• 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*}

  • Convergence: this is a formal stack, any \begin{align*}\[thickening\end{align*} ] are identified with something… On each finite approximation, most terms are \({\mathcal{O}}_X\).

• 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.

  • \begin{align*}\[topological periodic cyclic homology\end{align*} ] corresponds to \begin{align*}\[prismatic cohomology\end{align*} ]
    • \begin{align*}\[topological Hochschild homology\end{align*} ] corresponds to \begin{align*}\[Hodge-Tate cohomology\end{align*} ].

• Prismatic is filtered by things that look like Hodge-Tate

• Absolute = arithmetic (take eigenspaces, related to \begin{align*}\[motivic cohomology\end{align*} ], relative = geometric?

  • Link to \begin{align*}\[K-Theory\end{align*} ] comes from eigenspaces somehow.

• Similar to situation in \begin{align*}\[Étale cohomology\end{align*} ] : need absolute and relative to compute either.

Why are `

• Homological algebra lives in the \begin{align*}\[derived category\end{align*} ]
• In AG, tight link between \begin{align*}\[birational\end{align*} ] equivalence (of say smooth \begin{align*}\[projective varieties\end{align*} ] and equivalence of \({\mathsf{DCoh}}\), the derived categories of \begin{align*}\[coherent sheaves\end{align*} ]
• See the \begin{align*}\[Bondal-Orlov conjecture\end{align*} ].
  • \begin{align*}\[birational\end{align*} ] is a weakening of isomorphism between varieties
  • Being derived equivalent is a weakening of having equivalent \({\mathsf{DCoh}}\)
  • Both recover actual isomorphisms in the case of smooth projective varieties
• Rep theory: equivalent derived categories is called \begin{align*}\[Morita equivalence\end{align*} ].
  • Derived equivalence is a weakening of Morita equivalence
  • Can replace an algebra by a much simpler derived-equivalent one
  • Use to study \begin{align*}\[blocks\end{align*} ] of \begin{align*}\[group algebra\end{align*} ]
  • Lots of numerical consequences?

A Roadmap to `

Roadmap to HHR

Some `

Lurie’s Seminar on Algebraic Topology

A bunch of suggested papers

Lurie’s Topics in Geometric Topology

The Relationship Between `

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*} \(\mathop{\mathrm{Maps}}_{ \operatorname{DSt}}(S^1, X)\).

  • Points of \(LX\): infinitesimal loops in \(X\)

• Identify \({\operatorname{THH}}(X) \xrightarrow{\sim} {\mathcal{O}}(LX)\) (global functions)

  • Corollary of a result in Ben-Zvi–Francis–Nadler “Integral Transforms and Drinfeld Centers in \begin{align*}\[derived algebraic geometry\end{align*} ]”?

• \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 \(LX\)

  • Use the map \(LX\to X\) sending a loop to its basepoint

• 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

  • See Connes’ negative \begin{align*}\[cyclic homology\end{align*} ]

• Dennis trace is invariant under all covering maps of circles, even multisheeted

  • Encoded not in a group action by a \begin{align*}\[cyclotomic structure\end{align*} ].
  • Take homotopy fixed points of the cyclotomic structure on \({\operatorname{THH}}\) to get \({\operatorname{TC}}\), \begin{align*}\[Topological cyclic homology\end{align*} ]
  • There is a map \(K\to {\operatorname{TC}}\)

• Theorem of Dundas-Goodwillie-McCarthy: whenever \(A\to A'\) is a nilpotent extension of connective \begin{align*}\[ring spectra\end{align*} ], \(K(A') \xrightarrow{\sim} K(A) { \underset{\scriptscriptstyle {{\operatorname{TC}}(A)} }{\times} } {\operatorname{TC}}(A')\)

`

• Some good stuff from Akhil Mathew on EM spaces:
  • Blog post

Why Care About `

• Why should I care about stacks? #unanswered_questions

• Why should I care about \begin{align*}[derived stacks](#derived-stacks)\end{align*} ? #unanswered_questions

• 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*} ]

Homotopy Theory is Connected to `

`

• Definitions of schemes and scheme-y curves

• Definitions of schemes and scheme-y curves

• Grothendieck’s fundamental group

• Statement of class field theory in terms of fundamental groups

• 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?

2021-04-18

Notes on modular forms

#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.

  • A weight \(k\) modular form is an element of \(H^0(X; \omega^{\otimes k})\) where \(X\) is the compactified modular curve, a quotient of \(H \cup{\mathbb{P}}^1({\mathbb{Q}})\)
    • This definition extends to \(H/\Gamma\)

• Weird fact: \(M_1\) is one-dimensional, but for \(g\geq 2\) we have \(\dim M_g = 3g-3\)

  • Special things for \(g=1\): \(q {\hbox{-}}\)expansions (i.e. Fourier series), vanishing \begin{align*}\[Torelli\end{align*} ], \(\pi_1 {\mathbb{T}}\) for the torus is abelian, the \(\theta\) function has a discrete zero locus, infinite product expansions like Jacobi’s triple product

• 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.

  • People do care about \begin{align*}\[Siegel modular forms\end{align*} ] : replace \({\mathbb{H}}\) with \({\mathbb{H}}_S^g\) the symmetric \(g\times g\) matrices with positive-definite imaginary part

• \({\mathbb{H}}_S^g/{\mathsf{Sp}}(2g; {\mathbb{Z}})\) 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.

  • Surprisingly, \({\mathbb{H}}_S^g\) is a \begin{align*}\[Lie group\end{align*} ] but not a \begin{align*}\[Lie algebra\end{align*} ] : \([AB]^t = -[BA]^t\), so it’s not closed.

2021-04-17

The “Three Things” Exercise

#advice

Reminding myself of Ravi’s “Three Things” exercise.

\begin{align*}\[Three Things Exercise#Process for talks\end{align*} ]

2021-04-16

Stanford AG: Samir Canning, joint with Hannah Larson

#seminar_notes

• What is the \begin{align*}\[universal curve\end{align*} ] over \(M_g\)? #unanswered_questions

• What is the \begin{align*}\[tautological ring\end{align*} ], and how does it relate to the \begin{align*}\[Chow ring\end{align*} ]? #unanswered_questions

• What are the \begin{align*}\[kappa classes\end{align*} ]? #unanswered_questions

• 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.

2021-04-15

Beilinson-Bloch Conjecture

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?

  • Higher Chow groups: generalize the \begin{align*}\[Mordell-Weil group\end{align*} ] for \begin{align*}\[elliptic curve\end{align*} ]

• 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!