Most Popular
1500 questions
45
votes
4 answers
Transpositions of order three
Allow me to take advantage of your collective scholarliness...
The symmetric group $\mathbb S_n$ can be presented, as we all know, as the group freely generated by letters $\sigma_1,\dots,\sigma_{n-1}$ subject to…
Mariano Suárez-Álvarez
- 46,795
45
votes
1 answer
two tetrahedra in $\mathbb R^4$
It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $\mathbb R^3$,
such that their union has diameter $1,$ then they must share a vertex.
I wonder whether we have an analog of this in higher dimensions.…
filipm
- 1,359
45
votes
3 answers
Submission of a paper with a serious error to a good journal
I recently obtained (or so I thought) a good result, and after a month of reading and rereading what I'd written, submitted my paper to a very good journal. I'm early in my career (got my Ph.D. a few years ago) and have published 2 papers in good…
Bas Lag
- 1
45
votes
1 answer
Anti-concentration bound for permanents of Gaussian matrices?
In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i.i.d. Gaussian entries is "not too concentrated…
Scott Aaronson
- 9,743
45
votes
2 answers
"Closed-form" functions with half-exponential growth
Let's call a function f:N→N half-exponential if there exist constants 1
Scott Aaronson
- 9,743
45
votes
0 answers
A = B (but not quite); 3-d arrays with multiple recurrences
Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan)
1
1 3
2 10 15
6 40 105 105
24 196 700 1260 945
which is defined by $S(i,j) = i\ S(i-1,j) + (i+j)\ S(i-1,j-1)$ and $S(0,1)=1$, and…
Peter Shor
- 6,272
45
votes
13 answers
Motivating the de Rham theorem
In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book Foundations of Differentiable Manifolds and Lie Groups. One thing that I remember being puzzled by, and which I felt…
Timothy Chow
- 78,129
45
votes
7 answers
Good references for Rigged Hilbert spaces?
Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, even when the physicist is trying to be…
Todd Trimble
- 52,336
45
votes
5 answers
Calculating cup products using cellular cohomology
Most algebraic topology books (for instance, Hatcher) contain a recipe for computing cup products in singular or simplicial homology. In other words, given two explicit singular or simplicial cocycles, they contain a recipe for computing an…
DanT
- 451
45
votes
1 answer
Square roots of elements in a finite group and representation theory
Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that square to $g$, attains its maximum at the…
Alex B.
- 12,817
45
votes
1 answer
Consequences of Geometric Langlands
So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki because there isn't really a "right" answer: what…
Charles Siegel
- 15,805
45
votes
3 answers
Putnam 2020 inequality for complex numbers in the unit circle
The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020:
Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that $\left|z_i\right| \leq 1$ for each $i \in…
darij grinberg
- 33,095
45
votes
8 answers
A down-to-earth introduction to the uses of derived categories
When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about spectral sequences in order to use them.
Does…
Charles Staats
- 7,218
45
votes
2 answers
Applications of Zorn’s lemma that aren’t chain-complete/directed-complete?
Zorn’s Lemma applies to posets in which every chain has an upper bound. However, in all applications I know, the poset is also evidently chain-complete — chains have least upper bounds. A few classic such applications:
AC, using a poset of…
Peter LeFanu Lumsdaine
- 18,946
45
votes
2 answers
Can the fugitive escape?
A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively.
In a fugitive move, the fugitive can travel no more than a distance of $\delta$
In an officer move, the…
Eric
- 2,601