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: GaussBonnet and Hopf index theorem.
 GaussBonnet 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 : \({\mathbf{B}}S^1 = {\mathbf{CP}}^{\infty}\)
 Why are maximal tori useful?
What is a scheme?

https://www.ams.org/publications/journals/notices/201711/rnotip1300.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/AnelSemiomathsHomotopyColimit.pdf








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?
Tags: #personal/idlethoughts

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 rightexact, 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
ladic representations

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

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

See NéronOggShafarevich criterion: good reduction iff Inertia acts trivially, or semistable reduction iff inertia acts unipotently.

Always have quasiunipotently, 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.