All Questions
46,921
questions with no upvoted or accepted answers
213
votes
0
answers
16k
views
Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?
Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
- 2,181
138
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,387
106
votes
0
answers
16k
views
A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't ...
- 61.5k
92
votes
0
answers
16k
views
Hironaka's proof of resolution of singularities in positive characteristics
Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...
- 7,961
84
votes
0
answers
3k
views
Which finite abelian groups aren't homotopy groups of spheres?
Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know ...
- 21.5k
78
votes
0
answers
3k
views
The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p)...
- 4,745
77
votes
0
answers
4k
views
2, 3, and 4 (a possible fixed point result ?)
The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that
$$\Vert Tx-Ty\...
- 4,040
66
votes
0
answers
3k
views
Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1
Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
- 19.4k
62
votes
0
answers
2k
views
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 ...
- 115k
60
votes
1
answer
6k
views
Why "open immersion" rather than "open embedding"?
When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
57
votes
0
answers
3k
views
Grothendieck's Period Conjecture and the missing p-adic Hodge Theories
Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
- 137k
55
votes
0
answers
3k
views
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
54
votes
0
answers
2k
views
What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
- 1,228
54
votes
0
answers
2k
views
Uniformization over finite fields?
The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I ...
- 22.2k
53
votes
0
answers
2k
views
What is the current status of derived differential geometry?
I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many ...
- 2,035
52
votes
0
answers
992
views
Class function counting solutions of equation in finite group: when is it a virtual character?
Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
- 6,869
51
votes
0
answers
2k
views
Alternating colors on a line: infinitely often or converge?
Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
- 1,010
51
votes
0
answers
2k
views
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 ...
- 1,814
50
votes
0
answers
12k
views
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in ...
- 9,802
48
votes
0
answers
2k
views
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
- 38.2k
48
votes
0
answers
17k
views
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
- 2,984
47
votes
0
answers
2k
views
Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
- 42.1k
47
votes
0
answers
2k
views
Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositions: Can we prove there is no Borel decomposition?
This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of ...
- 225k
46
votes
0
answers
2k
views
Mikhalkin's tropical schemes versus Durov's tropical schemes
In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
- 6,169
46
votes
0
answers
6k
views
Cochains on Eilenberg-MacLane Spaces
Let $p$ be a prime number, let $k$ be a commutative ring in which $p=0$, and let
$X = K( {\mathbb Z}/p {\mathbb Z}, n)$ be an Eilenberg-MacLane space.
Let $F$ be the free $E_{\infty}$-algebra over $k$ ...
- 17.6k
45
votes
0
answers
2k
views
What is the "real" meaning of the $\hat A$ class (or the Todd class)?
In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...
- 6,666
45
votes
0
answers
910
views
Extending a line-arrangement so that the bounded components of its complement are triangles
Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that $\{L_1,\dots,L_m\}...
- 19.4k
45
votes
0
answers
1k
views
Enriched Categories: Ideals/Submodules and algebraic geometry
While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
- 3,161
45
votes
0
answers
3k
views
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
...
- 6,272
43
votes
0
answers
1k
views
Can a regular icosahedron contain a rational point on each face?
The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces?
For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, ...
- 22.2k
43
votes
0
answers
2k
views
Why are there so few quaternionic representations of simple groups?
Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
- 5,181
42
votes
0
answers
793
views
A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
- 40.9k
41
votes
0
answers
1k
views
Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?
Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
- 61.5k
41
votes
0
answers
2k
views
What does the theta divisor of a number field know about its arithmetic?
This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let ...
- 2,009
40
votes
0
answers
2k
views
Are we better in computing integrals than mathematicians of 19th century?
When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
40
votes
0
answers
1k
views
Homotopy type of TOP(4)/PL(4)
It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(...
- 6,139
40
votes
0
answers
3k
views
Minimal volume of 4-manifolds
This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
- 66.8k
40
votes
1
answer
5k
views
A generalisation of the equation $n = ab + ac + bc$
In a result I am currently studying (completely unrelated to number theory), I had to examine the solvability of the equation $n = ab+ac+bc$ where $n,a,b,c$ are positive integers $0 < a < b < ...
- 3,433
38
votes
0
answers
1k
views
Computer calculations in A_infinity categories?
Is there a good computer program for doing calculations in A-infinity categories?
Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep track ...
- 1,389
37
votes
0
answers
1k
views
Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
- 37.4k
37
votes
0
answers
5k
views
Homology of $\mathrm{PGL}_2(F)$
Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
- 19.9k
37
votes
0
answers
1k
views
Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
- 7,047
37
votes
0
answers
2k
views
What is the three-dimensional hyperbolic volume of a four-manifold?
Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
...
- 10.2k
36
votes
0
answers
867
views
Chern character of a Representation
Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-...
- 19.6k
36
votes
0
answers
1k
views
Rigid non-archimedean real closed fields
Question. Is there a countable rigid non-Archimedean real closed field?
Background:
As usual, a structure is said to be rigid if the only automorphism of the structure is the identity map.
It is ...
- 17.1k
36
votes
0
answers
930
views
Are there infinite versions of sporadic groups?
The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group.
For each of the groups of Lie ...
- 576
36
votes
0
answers
1k
views
3-colorings of the unit distance graph of $\Bbb R^3$
Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = (\alpha,...
- 16.3k
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 178k
36
votes
0
answers
1k
views
Functor that maps to both $KO^n$ and $KO^{-n}$
(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...
- 42.5k
36
votes
0
answers
1k
views
Two-convexity ⇒ Lefschetz?
Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...
- 43.7k