Видео

thanks for these valuable remarks!

@@manifoldsinmaryland Yup 👍. I think I found a better reference for why the affine line A^1 is used in the notes of “Motives I” by Eion Mackall. There near the end he mentions: “…the cohomology we work with satisfies the property H*(X x A^1) = H*(X). One way to see this is the calculation H^1(A^1) = 0 and then use the Künneth decomposition on the product. …we consider A^1 as a suitable substitute for the interval’s role in homotopy theory.” Thom’s theorem equates cobordism to a homotopy theory. Thom’s theorem derived that cobordisms groups are homotopy groups of (universal) Thom spectra MG by π_n(MG ⋀ X) = MG_n(X) with G a topological group such as the unitary group U. The associated cohomology theory of a spectra E is of interest considering the realization functor mentioned in 4 of Mackall. Whitehead showed that any homology is E -> π_(E ∧ X) = H_n(X; C(X)) for spectrum E on CW-complex X with coefficients C(X). Brown’s representation theorem or assuming Poincaré duality between homology and cohomology (ie Weil cohomology realized by motive of rational equivalence) it is possible to derive a cohomology theory from a spectrum E such as the Thom spectrum MG.

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

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

Glad you found our summary useful.

nevermind, it's because a closed segment isn't a manifold. which is stupid, but true nonetheless lol

True. Thanks for visiting the cobordism exhibition!

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

@@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!

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

too hard to learn ;p

one reason is that geometric algebra is considered even better way to rotate (but also somewhat harder to grasp), so why no learn it instead haha

Thank you so much 😀

Glad you liked it!

You're very welcome! Thanks for watching.

Glad to hear!

sounds like a good idea.

Thank you!

Thanks for noticing. X should be simply Omega here. Indeed, I should have said more about the construction of rho, but I am never really sure how much details is OK in math RUclips videos.

@@manifoldsinmaryland How can $X$ be $\Omega$ if the section $L\cap \Omega$ is small ? e.g. take $\Omega$ to be a ball in $\mathbb{C}^2$ and let $L=\{z_n =0\}$ be a complex line that does not pass through the center of $\Omega$ but intersects $\Omega$ in a nonempty open set. Then the formula $g(z_1,\dots, z_n)= f(z_1,\dots,z_{n-1})$ does not define $g$ on all of $\Omega$, but on a proper subset.

@@debraj10 not sure what you mean. I say specifically that g is defined on a proper open subset of Omega. Then I construct the global extension using a cutoff of L cap Omega.

For v, the limit of the v_i’s

indeed!

check back for more!

thanks for the suggestion!

@@manifoldsinmarylandI agree with Czeckie.a vid of Chern Moser normal forms would be fantastic!

thanks for the suggestion! I will put that on my list

We will!

Glad you enjoyed it!

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

@@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.

@@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 <0} (where l is some cohomology group in Hdg_ {2n}), my approach is that for the spaces $ P_ {l <0} - $ adic the preserved volumes (see it as an indicator of symmetry) are true for a subgroup of functions l g3 = 1, or if the P_ {l <0} \ to g3 = Mod 1 where well all the projections of a subgroup in 3 are relevant, in the volume P_ {l <0}, this case for example leads us to think of the previous idea as the "extension "of a semi-stable manifold $ \ log_ {q} $ must be identical to $ q (2n)> p $ geometrically representing the P-adic in this case, only as a volume of symmetries locally. This as the volume of symmetries locally is this maximum smooth subgroup g3 in a local finite Map_ {K} = (n-1). That is why I tell you that the P-adic space of a dense volume of symmetries is of intense understanding within the CH, and Hodge's own theory. So my general version of the, CH is the following consider whether $g3 \{ Mod |p, +1|\forall{} N_{|-1|}\} or g3= Mod|p, +1|\times{} \mathcal{N}_{|-1|} where see you that, the $\mathcal{N}_{|-1|}$ of an structure of submanifolds in $P_{*}$ (see you only as $\mathcal{N}\subset{} P_{*}$) is the standard case of, admitting P-adic in some N submanifold, Even since the P-addiction is of a soft or projective class, aEven since the P-addiction is of a soft or projective class, is traced a $P(adic)= P_{('(2n)+1)} but if $P_{*}\in{} N_{|-1|} then you written to $P(adic)= P_{(, (2n)-1)}$ or also written as $P(adic)= P_{l<0}$ where l are all dense and maximal correspondence functions, identical with modulo in a submanifold n-1, Proving that the CH is only true if g3 = Mod | p, +1 |. but not in 3 nor its present dimensions. I await your answers, if you want me to send you my paiper. greetings friend

100% a crank