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1500 questions
62
votes
4 answers
When size matters in category theory for the working mathematician
I think a related question might be this (Set-Theoretic Issues/Categories).
There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance Shulman - Set theory for category theory).
Some…
jg1896
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62
votes
68 answers
Mathematicians with both “very abstract” and “very applied” achievements
Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant interests that one wouldn’t guess one from the…
Francois Ziegler
- 29,500
62
votes
5 answers
Jean Bourgain's relatively lesser known significant contributions
Jean Bourgain passed away on December 22, 2018.
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions, such as the proof of Vinogradov's conjecture…
kodlu
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62
votes
2 answers
Non-abelian class field theory and fundamental groups
Over the years, I've been somewhat in the habit of asking questions in this vein to experts in the Langlands programme.
As is well known, given an algebraic number field $K$, they propose to replace the reciprocity map
$$A_K^\*/K^*\rightarrow…
Minhyong Kim
- 13,471
62
votes
9 answers
Can a vector space over an infinite field be a finite union of proper subspaces?
Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces?
If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are a finite number of 1-dimensional subspaces, and…
Anton Geraschenko
- 23,718
62
votes
4 answers
What is the limit of gcd(1! + 2! + ... + (n-1)! , n!) ?
Let $s_n = \sum_{i=1}^{n-1} i!$ and let $g_n = \gcd (s_n, n!)$. Then it is easy to see that $g_n$ divides $g_{n+1}$. The first few values of $g_n$, starting at $n=2$ are $1, 3, 3, 3, 9, 9, 9, 9, 9, 99$, where $g_{11}=99$. Then $g_n=99$ for $11\leq…
Robin Tucker-Drob
- 1,134
62
votes
14 answers
What advantage humans have over computers in mathematics?
Now that AlphaGo has just beaten Lee Sedol in Go and Deep Blue has beaten Garry Kasparov in chess in 1997, I wonder what advantage humans have over computers in mathematics?
More specifically, are there any fundamental reasons why a machine learning…
Māris Ozols
- 489
62
votes
19 answers
Generalizations of the four-color theorem
The four color theorem asserts that every planar graph can be properly colored by four colors.
The purpose of this question is to collect generalizations, variations, and strengthenings of the four color theorem with a description of their status.…
Gil Kalai
- 24,218
62
votes
22 answers
What are the worst notations, in your opinion?
With which notation do you feel uncomfortable?
user2330
- 1,310
62
votes
0 answers
Are there periodicity phenomena in manifold topology with odd period?
The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds have Euler characteristic zero, while…
Qiaochu Yuan
- 114,941
62
votes
9 answers
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology is that if $S$ is a noncompact connected surface,…
Andy Putman
- 43,430
62
votes
5 answers
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can choose generators for M however I please, and still…
Andrew Critch
- 11,060
62
votes
25 answers
Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to the introduction of abstract vector spaces mid-way…
Dan Ramras
- 8,498
62
votes
11 answers
Why is the exterior algebra so ubiquitous?
The exterior algebra of a vector space V seems to appear all over the place, such as in
the definition of the cross product and determinant,
the description of the Grassmannian as a variety,
the description of irreducible representations of…
Qiaochu Yuan
- 114,941
62
votes
22 answers
What's a groupoid? What's a good example of a groupoid?
Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?
Emily Peters
- 1,089