Michael Hopkins: Bernoulli numbers, homotopy groups, and Milnor

Abstract:
In his address at the 1958 International Congress of Mathematicians Milnor described his joint work with Kervaire, relating Bernoulli numbers, homotopy groups, and the theory of manifolds. These ideas soon led them to one of the most remarkable formulas in mathematics, relating fundamental quantities from three different fields. This talk will describe this formula, and the many remarkable developments associated to each of its terms.
This lecture was held at The University of Oslo, May 25, 2011 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.
Program for the Abel Lectures 2011
1. "Spheres" by Abel Laureate John Milnor, Institute for Mathematical Sciences, Stony Brook University, New York
2. "Manifolds, topology and dynamic" by Professor Curtis McMullen
3. "Bernoulli numbers, homotopy groups, and Milnor" by professor Michael Hopkins
4. "A guided tour of the seventh dimension", a science lecture by professor Etienne Ghys

[@giovannysoto4151]

This was an amazing lecture!

[@NoNTr1v1aL]

Amazing video!

[@gooomaaal]

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]

🤔☕

[@TIENTI0000]

excellent