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1500 questions
51
votes
3 answers
Spaces with same homotopy and homology groups that are not homotopy equivalent?
A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are there two non-homotopy equivalent spaces with…
Dylan Thurston
- 10,033
51
votes
8 answers
Is there a fast way to check if a matrix has any small eigenvalues?
I have hundreds of millions of symmetric 0/1-matrices of moderate size (say 20x20 to 30x30) which (obviously) have real eigenvalues.
I wish to extract from this list the tiny number of matrices that have no small eigenvalues, where I am calling an…
Gordon Royle
- 12,288
51
votes
24 answers
Most elementary proof showing that exponential growth wins against polynomial growth
This question is motivated by teaching : I would like to see a completely elementary proof showing for example that for all natural integers $k$ we have eventually $2^n>n^k$.
All proofs I know rely somehow on properties of the logarithm.
(I have…
Roland Bacher
- 17,432
51
votes
7 answers
Does anyone still seriously doubt the consistency of $ZFC$?
As someone self-taught in set theory beginning with Donald Monk’s excellent book on MK set theory, $ZFC$ has always seemed like a weak set theory.
Despite this, the majority of professional contemporary mathematicians still seem to view it as a very…
Alec Rhea
- 9,009
51
votes
14 answers
Introductory text on geometric group theory?
Can someone indicate me a good introductory text on geometric group theory?
Gian Maria Dall'Ara
- 2,449
51
votes
2 answers
The "square root" of a graph?
The number $f(n)$ of graphs on the vertex set $\{1,\dots,n\}$,
allowing loops but not multiple edges, is $2^{{n+1\choose
2}}$, with exponential generating function $F(x)=\sum_{n\geq 0}
2^{{n+1\choose 2}}\frac{x^n}{n!}$. Consider
$$ \sqrt{F(x)} =…
Richard Stanley
- 49,238
51
votes
4 answers
How hard is it to compute the number of prime factors of a given integer?
I asked a related question on this mathoverflow thread. That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered.
So quoting myself from that thread:
How hard is…
Rune
- 2,386
51
votes
0 answers
Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?
A very specific case of Reed's Conjecture
Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic number, $\Delta$ is the maximum degree, and…
Andrew D. King
- 1,814
51
votes
1 answer
Is there a notion of polynomial ring in "one half variable"?
Let $C$ be the category of commutative rings.
Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ?
(Here, we may assume those isomorphisms to be natural in $R$, if needed).
I tried to see what $F(\mathbb…
Watson
- 1,702
51
votes
6 answers
How is representation theory used in modular/automorphic forms?
There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and representation theory?
I have a basic grounding in the…
David Corwin
- 15,078
51
votes
2 answers
What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as in Higher Topos Theory (HTT) 6.1.0.4: an…
Charles Rezk
- 26,634
51
votes
4 answers
A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?
Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.
Of course, Lebesgue and Poincaré knew each other, they even met on several occasions and shared a common close friend, Émile…
Fabrice Pautot
- 1,016
51
votes
9 answers
Is Galois theory necessary (in a basic graduate algebra course)?
By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) year graduate students who are usually attracted…
Jim Humphreys
- 52,369
51
votes
4 answers
Function satisfying $f^{-1} =f'$
How many functions are there which are differentiable on $(0,\infty)$ and that satisfy the relation $f^{-1}=f'$?
C.S.
- 4,735
51
votes
22 answers
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the following aim: I want to give the students a concrete…
Menny
- 638