Michael Hopkins: Bernoulli numbers, homotopy groups, and Milnor

[@giovannysoto4151]

This was an amazing lecture!

[@NoNTr1v1aL]

Amazing video!

[@TIENTI0000]

excellent

[@dupont7945]

Anyone can tell me how did he make the sphere to a torus (start from around 4:00)?😹

[@ChronusZed]

It's a magic trick. Probably when he pulled out the handkerchief to grab the first balloon off the floor he simultaneously pulled a second balloon out of his pocket and swapped the two while bent over.

[@hixidom2274]

What's the difference between "smooth transformation", "h-cobordism", and "diffeomorphism"?

[@akrishna1729]

so manifolds are spaces that are locally similar to some Euclidean space. formally, we can take patches ("charts") on a manifold and relay them via functions to certain subsets of R^n. the regularity of the transition maps dictates the kind of structure that our manifold has. a smooth (or C^\infty) map is simply one which is continuously differentiable of all orders (infinitely so); a diffeomorphism is a bijective smooth function with smooth inverse. a smooth manifold is a (topological) manifold whose local maps to Euclidean space are diffeomorphisms, i.e. preserving smooth structure. now let's deal with smooth manifolds, say, M and N of dimension k. a cobordism between M and N is a compact (k+1)-dimensional manifold W whose boundary consists of the disjoint union of M and N. h-cobordism is a special case of general cobordant M, N as above. the "h" stands for homotopy, what we can think of as some kind of deformation. in a cobordism situation (W; M, N), we have M and N are h-cobordant wrt W if the tautological inclusion maps from both M and N to W are homotopy equivalences, distinguishing these as especially "good" or controlled cases of cobordism. i know this is a late response, but hopefully it helps anyone who does come across it.

[@mambu6]

Who is jack ^^

[@EricWeinsteinPhD]

John Milnor.

[@MrArmas555]

+++

[@beyalexander2786]

🤔☕

[@gooomaaal]

excellent