# 2021-04-23

## What is equivariant cohomology?

• 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: • What is the Homotopy quotient? #todo/questions

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

• • Is this not precisely the Borel construction? #todo/questions

• classifying spaces : $${\mathbf{B}}S^1 = {\mathbb{CP}}^{\infty}$$   • Why are maximal torii useful?
• #todo/questions

## What is a scheme?

• 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

• • • • • • • • 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! • 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?

• 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

• 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.