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1500 questions
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votes
9 answers

What are "classical groups"?

Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with Weyl's book The Classical Groups but doesn't…
Jim Humphreys
  • 52,369
70
votes
6 answers

third stable homotopy group of spheres via geometry?

It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a generator of $\pi_8 (S^5)=\pi_{3}^{st}$. It is…
Johannes Ebert
  • 20,634
70
votes
6 answers

What are Jacob Lurie's key insights?

This question is inspired by this Tim Gowers blogpost. I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key insight allowed him to begin his programme and…
Daniel Moskovich
  • 21,887
70
votes
22 answers

Small ideas that became big

I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research opening new paths or suggesting new ways of thinking…
Hvjurthuk
  • 573
70
votes
10 answers

The Planck constant for mathematicians

The questions Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant? Q2. How does the Planck constant appear in mathematics of quantum mechanics? In particular, quantization is an important notion in…
Gil Kalai
  • 24,218
70
votes
5 answers

Does anyone know a polynomial whose lack of roots can't be proved?

In Ebbinghaus-Flum-Thomas's Introduction to Mathematical Logic, the following assertion is made: If ZFC is consistent, then one can obtain a polynomial $P(x_1, ..., x_n)$ which has no roots in the integers. However, this cannot be proved (within…
Akhil Mathew
  • 25,291
70
votes
2 answers

Group cohomology and condensed matter

I am mystified by formulas that I find in the condensed matter literature (see Symmetry protected topological orders and the group cohomology of their symmetry group arXiv:1106.4772v6 (pdf) by Chen, Gu, Liu, and Wen). These formulas have been used…
Edward Witten
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70
votes
7 answers

Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known from the beginning that it would take experts…
Ovi
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70
votes
2 answers

What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?

One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in commutative algebra, and really sounds as though it…
Ben Webster
  • 43,949
70
votes
9 answers

How does one find out what's happening in contemporary mathematics research?

How does one find out what's happening in contemporary mathematics research? EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside academia for quite a few years, and the gap between…
Michael
  • 2,175
70
votes
14 answers

Elementary / Interesting proofs of the Nullstellensatz

Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques? One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals correspond to points (using normalization), and then…
Abhishek Parab
  • 879
70
votes
9 answers

Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders $a_i$ where $a_i|a_{i+1}$ and a free abelian…
Ben Webster
  • 43,949
70
votes
30 answers

What programming languages do mathematicians use?

I understand this might be a slightly subjective question, but I am honestly curious what programming languages are used by the mathematics community. I would imagine that there is a group of mathematicians out there that use haskell because it…
user3063
  • 111
69
votes
19 answers

What are some results in mathematics that have snappy proofs using model theory?

I am preparing to teach a short course on "applied model theory" at UGA this summer. To draw people in, I am looking to create a BIG LIST of results in mathematics that have nice proofs using model theory. (I do not require that model theory be…
Pete L. Clark
  • 64,763
69
votes
7 answers

The main theorems of category theory and their applications

This question first arose as I wrote an answer for the question: Is there a nice application of category theory to functional/complex/harmonic analysis?; it can also be regarded as a (hopefully) more focused version of the question Most striking…
Paul Siegel
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