2021-04-23

What is equivariant cohomology?

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

• Some uses:

  • Calculate number of rational curve in a quintic threefold (Kontsevich 1995)

  • Calculate characteristic numbers of a compact homogeneous space (Tu 2010)

  • Derive Gysin formula for examples of fiber bundles whose fibers are homogeneous spaces (Tu 2011)

  • Calculate integrals over manifolds as sums over fixed points: Gauss-Bonnet and Hopf index theorem.

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

• What is the Homotopy quotient? #unanswered_questions

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

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  • Is this not precisely the Borel construction? #unanswered_questions

• classifying spaces : \({\mathbf{B}}S^1 = {\mathbb{CP}}^{\infty}\)

• Why are maximal torii 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, scheme are the place to do algebra by finding zeros of functions.

• Closed point : of the form \(V(S) \coloneqq\left\{{ q\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}q\supseteq S}\right\}\)

Notes on homotopy colimit via Diagrams

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

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• Hocolims are infinity groupoids, equivalently homotopy type.

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

15:07

• Hironaka: Fields for existence of Resolution of singularities in every dimension in \(\operatorname{ch}(k) = 0\).

Time Management

• Setting goals: SMART. Doesn’t work for research though!

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• Make lists, and habitually review/revise/plan.

• I really like the “keeping a problem list” idea.

• Don’t be ashamed to ask people if they have problems you can work on.

group cohomology in homotopy theory?

Tags: #idle_thoughts

• Thinking about the link between group cohomology and homotopy theory: if I have a SES \begin{align*} 0\to A \to B \to C \to 0 ,\end{align*} should one apply a functor like \(K({-}, 1)\)? Is this actually a functor…? We definitely get spaces \(K(A, 1)\) and \(K(B, 1)\), for example, and there must be an induced map between them. Want to make precise what it means to get a SES like this: \begin{align*} 0 \to K(A, 1) \to K(B, 1) \to K(C, 1) \to 0 .\end{align*} One would kind of want this to be part of a fiber sequence I guess. But we’re in \({\mathsf{Top}}\) anyway, so there’s no real issue with just doing fibrant and cofibrant objects,.

  Maybe the “right” think to do here is to actually take a classifying groupoid (?), 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

l-adic representations

• Try computing things like \({ \mathsf{Gal}} ({\mathbb{Q}}( \zeta_3, \sqrt{3})\).

• There’s some way to check orders of Galois groups using valuation..?

• See Néron-Ogg-Shafarevich criterion: good reduction iff Inertia acts trivially, or semistable reduction iff inertia acts unipotently.

• Always have quasi-unipotently, so eigenvalues roots of unity.

  • Easy for elliptic curve.
  • For moduli stack of abelian varieties, requires Néron models, see Silverman.

• Galois representations at different primes are related, using local info at a few primes to get global info at all primes.

17:13

• Relation between quadratic form and unique factorization:

22:29

• See Marcus (?) for a nice proof of quadratic reciprocity involving looking at primes splitting in quadratic fields.