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user 613
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
21
votes
What are some famous rejections of correct mathematics?
What about deBranges' proof of the Bieberbach conjecture:
https://en.wikipedia.org/wiki/Louis_de_Branges_de_Bourcia
- 4,713
2
votes
Intuition/meaning behind/physical content of the concept of a smooth structure
$\newcommand{\R}{\mathbb{R}}$The purpose of a manifold structure is to be able to differentiate functions. And initially we know how to differentiate functions only if the domain is an open set in $\R …
- 12.9k
98
votes
What's a great christmas present for someone with a PhD in Mathematics?
I'm surprised no one has yet suggested a lifetime supply of Hagoromo chalk.
- 27.5k
5
votes
Prominent non-mathematical work of mathematicians
Although John Urschel is already mentioned in a comment to the answer about Frank Ryan, I think he deserves his own answer. Urschel was a football player first at Penn State and then with the Baltimor …
- 27.5k
9
votes
What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed t...
I consider the distinction quite important. There are two separate operations that look superficially like each other but are in fact different.
First, the abstract description. If $V$ is an abstract …
- 27.5k
2
votes
Famous conjectures named after a mathematician that were resolved in their lifetimes
Thurston was still alive, when Perelman solved the Thurston conjecture.
- 27.5k
59
votes
Examples of simultaneous independent breakthroughs
One of the most entertaining seminar talks I ever attended was by Michael Atiyah on the moment map on a Lie group. As the talk progressed, the people in the front row became more and more agitated, un …
- 27.5k
6
votes
Accepted
Hyperbolic PDE in mathematics
Hyperbolic PDEs arise unexpectedly in some differential geometric questions involving prescribed data. What's weird in these cases is that there is no natural time coordinate in the PDEs. Here are som …
- 27.5k
5
votes
ICM 2018 lecture videos
There are now several more videos of ICM talks available. I imagine that even more will be posted in the near future.
https://www.youtube.com/channel/UCnMLdlOoLICBNcEzjMLOc7w/videos
- 27.5k
10
votes
Examples of "miraculous" proofs
It's not really a theorem and maybe not miraculous, but I think the way Gromov, in his book Partial Differential Relations solves an underdetermined system of PDEs (fewer equations than unknown functi …
- 27.5k
5
votes
Examples of "miraculous" proofs
Gromov's theorem on groups with polynomial growth has, I think, an amazing proof. It's where he introduced Gromov-Hausdorff distance and showed, using the solution to Hilbert's fifth problem, that a …
- 27.5k
7
votes
Examples of "miraculous" proofs
My favorite example is Gunther’s radically simpler proof of the Nash isometric embedding theorem, which came out of nowhere.
- 27.5k
14
votes
What math institutes offer research in pairs/research in teams?
The American Institute of Mathematics in San Jose has a nice program for collaborations:
http://aimath.org/research/squares.html
- 28.8k
2
votes
Accepted
Automatic transfer of pointwise metric computations to bundle computations
The following is the meta-theorem I have in mind (I can't swear that what I've written is 100% correct):
Given $G < \mathrm{GL}(n)$, let $\Phi: \mathbb{R}^n \rightarrow \mathbb{R}$ a smooth $G$-invar …
- 27.5k
1
vote
Automatic transfer of pointwise metric computations to bundle computations
I think everything can be reduced to the following (stated without proofs):
Let $E$ and $F$ be rank $n$ oriented vector bundles over $\mathcal{M}$ with smooth inner products and compatible connection …
- 27.5k