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 Unsorted/list of fibrations 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? #todo/questions

    • attachments/image-20210218013730610.png
  • If \(G\curvearrowright M\) with \(G\) a compact connected Lie group, Cartan constructs a chain complex from \(M, {\mathfrak{g}}\).

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

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What is a scheme?

Notes on homotopy colimit via Diagrams

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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!

    attachments/image_2021-04-23-15-54-19.png

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

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  • 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: #personal/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:

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22:29

  • See Marcus (?) for a nice proof of quadratic reciprocity involving looking at primes splitting in quadratic fields.
#web/quick-notes #todo/questions #personal/idle-thoughts