What is Hodge theory?

Summary of Hodge theory on compact Riemannian manifolds, with Poincare duality as an application

[@NoNTr1v1aL]

Amazing video! Subscribed.

[@manifoldsinmaryland]

Thanks for the sub!

[@teamoverc-hellletloose]

at 23:30, the star of the poincare duality and the use of Hodge star operator are used in the same place, kind of confusing.

[@manifoldsinmaryland]

true, I wish there was something I could do about it :)

[@teamoverc-hellletloose]

@@manifoldsinmaryland Anyway, great structral explaination and thank you for the video! I also did a talk on the Hodge Theory on Monday :)

[@ficsur86]

Glad you found our summary useful.

[@alexboche1349]

What background do I need to understand this? Learning differenential forms now. I know very basic stuff about manifolds e.g. tangent space/bundle. I don't know differential geometry.

[@manifoldsinmaryland]

That should be enough to get started. A good book that is suitable for your background is Wells' Differential Analysis on complex manifolds

[@alexboche1349]

@@manifoldsinmaryland I didn't know if you would answer so I didn't give that much info. Let me give a little more. I want to understand the generalization of Helmholtz decomposition to n dimensions. Wikipedia says that's called Hodge theory. I don't think I need too much about *complex* manifolds specifically but maybe that helps. I'm from economics / game theory not physics/math. Does your book recommendation still stand? OR is there a better book in that case? Thanks!

[@manifoldsinmaryland]

@@alexboche1349 It does not stand anymore. Unfortuantely, I know little to nothing about Hemholtz equations :(

[@cristhiangalindo4800]

Good can you make a talk, about the Hodge conjecture, I am working the CH for example for a dimension p-addic "convergences-thus falls" where the CH is true only under a 2-cycle Hodge, because I found that p- adic ($P= N_{|-1|}$ as an isomorfism of the algebraic in $N\subset{} K$) where the $Map_{K}=1$ an also this contains to $Map_{K}: \sum_{j}^{\infty= - i}$ where the P-adic case on the dimensión $q(2n)> p$ it is well a complement with the subvariety (n-1) where the contains "symmetric" is localy finite in $\delta\mathfrak{H} \to (\delta, - \delta)$ (which takes for a finite thing, its value in the regular Homotopia), Where if the Homotopy coincides with Mod p (1), then a discrete set of normal walls is produce.

[@manifoldsinmaryland]

Thanks for the comment. Unfortunately I know very little about algebraic geometry over the p-adics.

[@cristhiangalindo4800]

@@manifoldsinmaryland Yes, of course, but knowing things about what a 2-cycle is, for example from Hodge, your P-adic space can be an application of how this is continuously pasted into a complex N_ {| -1 |} submanifold.

[@cristhiangalindo4800]

@@manifoldsinmaryland For example, you can consider the space P-adic (which in general is $Q^{*}(H)$-adic or also as $Q^{*+n}$-adic of cycle ) On some standard volume (group SU3 in the hyperplane convex) , which corresponds to a Hodge-structure, I for example have been studying the P-adic for Hodge- 2-dimensional structures, like $ P: Hdg_ {2} here you see that also P \ xrightarrow {\ \ cong \} Hdg_ {2}, where well any P-adic volume of a Hodge-structure is identical to an isomorphism -linear, Thanks to this dear colleague, it has been possible to generalize the P-adic volume (see it as a Module-potent oin SU_{Z3} =Mod-p^{2} (x;i)) where all the Q(H,n)-cycles of structure linear are compatible with the linear part of some P-adic volume, from here for example the Hodge-conjecture arises but in this case for example the P-adic is wrapped in P = | -1 |\ in {} N (see N as the submanifolds of N_{g^{p,q}}$) where the P-adic corresponds to a Hodge structure of class q (2n)> p (case of 2 Hodge cycles), where well the volume of all Hodge structures is generated in a semi-stable abelian manifold, see how P (adc) = N_ {| -1 |} \ to \ log_ {q} (where \ log_ {q} = \ log_ {, (2n) - xi}) which is where all logarithmic abelian manifolds admit some volume "in q" of the structures of 2 -Hodge cycle, for example, you can write to q: = P * = P (adic). Currently, I am working on a paiper, (which if you want I can send you), with the CH (conjecture-Hodge), for P-adic which are associated functions of P_ {l

[@98danielray]

100% a crank