Most Popular
1500 questions
65
votes
17 answers
Good introductory references on algebraic stacks?
Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also browsed through FGA explained (Fantechi et al.).…
Daniel Bergh
- 1,538
65
votes
9 answers
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
Thanh
- 651
65
votes
1 answer
Did Bourbaki write a text on algebraic geometry?
Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?
Georges Elencwajg
- 46,833
65
votes
3 answers
How many unit cylinders can touch a unit ball?
What is the maximum number $k$ of unit radius, infinitely long cylinders with mutually disjoint interiors that can touch a unit ball?
By a cylinder I mean a set congruent to the Cartesian product of a line and a circular disk. The illustrations,…
Wlodek Kuperberg
- 7,256
65
votes
3 answers
Reasons to prefer one large prime over another to approximate characteristic zero
Background:
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather than over $\mathbb Q$. (Note that working…
Charles Staats
- 7,218
64
votes
5 answers
Why tropical geometry?
Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being given by the (usual) maximum of real numbers and…
Robert Kucharczyk
- 1,288
64
votes
6 answers
Shortest closed curve to inspect a sphere
Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and
exterior to $S$
which has the property that every point $x$ on $S$ is visible to some point $y$ of $C$,
in the sense that the segment $xy$…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
64
votes
6 answers
Origin of terms "flag", "flag manifold", "flag variety"?
These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably the concrete notion of (complete) flag of…
Jim Humphreys
- 52,369
64
votes
2 answers
Stiefel–Whitney classes in the spirit of Chern-Weil
Chern-Weil theory gives characteristic classes (e.g. Chern class, Euler class, Pontryagin) of a vector bundle in terms of polynomials in the curvature form of an arbitrary connection. There seems to be no hope in getting Stiefel-Whitney classes…
Eric O. Korman
- 3,204
64
votes
1 answer
Is there a "classical" proof of this $j$-value congruence?
Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{C}/(\mathbf{Z} + \mathbf{Z}\tau)$, so this has…
BCnrd
- 6,978
64
votes
6 answers
What is the simplest proof that the density of primes goes to zero?
By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$.
I want something much weaker, namely that the proportion just goes to zero, at…
Kim
- 4,084
64
votes
2 answers
Who is the "young student" André Weil is referring to in his letter from the prison?
I am reading a nice booklet (in Italian) containing the exchange of letters that André and Simone Weil had in 1940, when André was in Rouen prison for having refused to accomplish his military duties.
Of course, among these letters, there is the…
Francesco Polizzi
- 65,122
64
votes
1 answer
J. H. C Whitehead (and his pig)
I am actually completing Master's Thesis on Lawson Homology. In order to do this, I am writing an appendix on Higher Homotopy Groups. Now, as you know, one of the most important Homotopy theorists ever is J. H. C. Whitehead and much of the standard…
Vincenzo Zaccaro
- 1,252
- 12
- 20
64
votes
1 answer
How to be rigorous about combinatorial algorithms?
1. The question
This may be the worst question I've ever posed on MathOverflow: broad,
open-ended and likely to produce heat. Yet, I think any progress that will be
made here will be extremely useful to the whole subject. In a nutshell, the
question…
darij grinberg
- 33,095
64
votes
3 answers
Chebyshev polynomials of the first kind and primality testing
Can you provide a proof or a counterexample for the claim given below ?
Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim :
Let $n$ be a natural number greater than two . Let $r$ be…
Pedja
- 2,673