--- date: 2021-04-23 tags: [ web/quick-notes ] --- # 2021-04-23 ## What is [equivariant cohomology](equivariant%20cohomology.md)? > - Some uses: - Calculate number of [rational curve](rational%20curve) in a [quintic](quintic) [threefold](threefold) (Kontsevich 1995) - Calculate characteristic numbers of a compact [homogeneous space](homogeneous%20space.md) (Tu 2010) - Derive [Gysin formula](Gysin%20formula) for [Unsorted/list of fibrations](Unsorted/list%20of%20fibrations.md) whose fibers are homogeneous spaces (Tu 2011) - Calculate integrals over manifolds as sums over fixed points: [Gauss-Bonnet](Gauss-Bonnet) and [Hopf index theorem](Hopf%20index%20theorem). - Gauss-Bonnet and Hopf index theorem: ![](attachments/image-20210218021511916.png) - What is the [Homotopy quotient](Homotopy%20quotient)? #todo/questions - ![](attachments/image-20210218013730610.png) - If $G\actson M$ with $G$ a compact connected [Lie group](Lie%20group.md), Cartan constructs a chain complex from $M, \lieg$. - ![](attachments/image-20210218013828220.png) - Is this not precisely the [Borel construction](Borel%20construction)? #todo/questions - [classifying spaces](classifying%20spaces) : $\B S^1 = \CP^{\infty}$ ![](attachments/image-20210218014024499.png) ![](attachments/image-20210218014113744.png) ![](attachments/image-20210218014335149.png) - Why are [[maximal tori]] useful? - ![](attachments/image-20210218014526989.png) #todo/questions ## What is a scheme? - - Manifolds are the place to do differential calculus, [scheme](scheme.md) are the place to do algebra by finding zeros of functions. - [Closed point](Closed%20point) : of the form $V(S) \da \ts{ q\in \spec R \st q\contains S}$ ## Notes on [homotopy colimit](homotopy%20colimit) via Diagrams - - ![](attachments/image-20210218010308802.png) - ![](attachments/image-20210218010627345.png) - ![](attachments/image-20210218010639958.png) - ![](attachments/image-20210218010749422.png) - ![](attachments/image-20210218010809139.png) - ![](attachments/image-20210218011118003.png) - ![](attachments/image-20210218012454677.png) - Hocolims are [infinity groupoids](infinity%20groupoids.md), equivalently [homotopy type](homotopy%20type.md). - There is a functor $\pi_0: \inftyGrpd\to \Set$. ![](attachments/image-20210218013131244.png) ## 15:07 - Hironaka: Fields for existence of [Resolution of singularities](Resolution%20of%20singularities.md) in every dimension in $\ch(k) = 0$. ![](2021-04-23%20Advice%20on%20research%20and%20problems) ## 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. ![](attachments/image_2021-04-23-15-55-40.png) - 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](group%20cohomology.md) in [homotopy theory](homotopy%20theory.md)? Tags: #personal/idle-thoughts - Thinking about the link between group cohomology and homotopy theory: if I have a SES \[ 0\to A \to B \to C \to 0 ,\] should one apply a functor like $K(\wait, 1)$? 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: \[ 0 \to K(A, 1) \to K(B, 1) \to K(C, 1) \to 0 .\] One would kind of want this to be part of a [fiber sequence](fiber%20sequence) I guess. But we're in $\Top$ anyway, so there's no real issue with just doing [fibrant and cofibrant objects](fibrant%20and%20cofibrant%20objects.md),. Maybe the "right" think to do here is to actually take a classifying [groupoid](groupoid.md) (?), which must be some functor like $\B: \Grp \to \Grpd$. Surely this is some known thing. But then what is an "exact sequence of groupoids"...? \[ 0 \to \B A \to \B B \to \B C \to 0 .\] 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 $\Ld \B(\wait)$? I think this just needs a model category structure on the *source*, although it seems reasonable to expect that $\Grpd$ would have some simple model structure. ## SeZoom [l-adic representations](l-adic%20representations.md) - Try computing things like $\Gal (\QQ( \zeta_3, \sqrt{3})$. - There's some way to check orders of Galois groups using [valuation](valuation)..? - See Néron-Ogg-Shafarevich criterion: [good reduction](good%20reduction.md) iff [Inertia](Inertia) acts trivially, or [semistable reduction](semistable%20reduction.md) iff inertia acts [unipotently](unipotently). - Always have quasi-unipotently, so eigenvalues roots of unity. - Easy for [elliptic curve](elliptic%20curve.md). - For [moduli stack of abelian varieties](moduli%20stack%20of%20abelian%20varieties), requires [Néron models](Neron%20model), see Silverman. - [Galois representations](Galois%20representations.md) 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](quadratic%20form.md) and unique factorization: ![](attachments/image_2021-04-23-17-13-44.png) ![](attachments/image_2021-04-23-17-14-05.png) ![](attachments/image_2021-04-23-17-14-22.png) ## 22:29 - See Marcus (?) for a nice proof of [quadratic reciprocity](quadratic%20reciprocity.md) involving looking at primes splitting in [quadratic fields](quadratic%20fields).