What is equivariant cohomology?
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Some uses:
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Calculate number of rational curve in a quintic threefold (Kontsevich 1995)
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Calculate characteristic numbers of a compact homogeneous space (Tu 2010)
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Derive Gysin formula for Unsorted/list of fibrations whose fibers are homogeneous spaces (Tu 2011)
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Calculate integrals over manifolds as sums over fixed points: Gauss-Bonnet and Hopf index theorem.
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Gauss-Bonnet and Hopf index theorem:
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Gauss-Bonnet and Hopf index theorem:
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What is the Homotopy quotient? #todo/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? #todo/questions
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classifying spaces : \({\mathbf{B}}S^1 = {\mathbf{CP}}^{\infty}\)
- Why are maximal tori useful?
What is a scheme?
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https://www.ams.org/publications/journals/notices/201711/rnoti-p1300.pdf
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Manifolds are the place to do differential calculus, scheme are the place to do algebra by finding zeros of functions.
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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
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http://mathieu.anel.free.fr/mat/doc/Anel-Semiomaths-HomotopyColimit.pdf
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Hocolims are infinity groupoids, equivalently homotopy type.
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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
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Setting goals: SMART. Doesn’t work for research though!
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Make lists, and habitually review/revise/plan.
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I really like the “keeping a problem list” idea.
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Don’t be ashamed to ask people if they have problems you can work on.
group cohomology in homotopy theory?
Tags: #personal/idle-thoughts
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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
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Try computing things like \({ \mathsf{Gal}} ({\mathbf{Q}}( \zeta_3, \sqrt{3})\).
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There’s some way to check orders of Galois groups using valuation..?
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See Néron-Ogg-Shafarevich criterion: good reduction iff Inertia acts trivially, or semistable reduction iff inertia acts unipotently.
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Always have quasi-unipotently, so eigenvalues roots of unity.
- Easy for elliptic curve.
- For moduli stack of abelian varieties, requires Néron models, see Silverman.
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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.