18
votes
0answers
958 views

Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$

A while ago, I came across the following problem, which I was not able to resolve one way or the other. Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and ...
9
votes
3answers
508 views

Embedding groups into groups with some vanishing homology groups

Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H_i(\Gamma;\mathbb{Z})=0$ for all $i \in ...
3
votes
1answer
509 views

When can a family of polynomials get a weight function to be made orthogonal?

Let $\lbrace P_n(z)\rbrace_{n\in\mathbb N_0}$ be a family of polynomials defined by a generating function $g(t,z)=\sum\limits_{n=0}^\infty P_n(z)t^n$ or by a contour integral $P_n(z)=\frac1{2\pi ...
12
votes
0answers
664 views

How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...
1
vote
3answers
5k views

Cycle of length 4 in an undirected graph

Can anyone give me a hint for an algorithm to find a simple cycle of length 4 (4 edges and 4 vertices that is) in an undirected graph, given as an adjacency list? It needs to use $O(v^3)$ operations ...
2
votes
3answers
1k views

Universal property of blowups

Can anyone help me with a proof of the following claim (see for example the book Higher algebraic geometry of Olivier Debarre, proof of Proposition 1.43, page 31): Let X be a complex manifold, and ...
16
votes
1answer
2k views

А generalization of Gromov's theorem on polynomial growth

I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao). Assume for a group $G$ there is a polynomial $P$ such that given $n\in\mathbb N$ there is set of ...
8
votes
2answers
376 views

Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...
11
votes
2answers
877 views

The maximum order of finite subgroups in $GL(n,Q)$

For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders. It appear in many articles a ...
5
votes
3answers
3k views

Non-negative integer solutions of a single Linear Diophantine Equation

Consider the following linear Diophantine Equation:: ax + by + cz = d ------------ (1) for all, a,b,c and d positive integers, and relatively prime, and ...
13
votes
1answer
404 views

Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...
4
votes
2answers
713 views

Special divisors on hyperelliptic curves

I was reading a proof that used the following result Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree ...
6
votes
1answer
587 views

Commuting Grothendieck topologies.

Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 ∪ T_2$ (by which I mean the smallest Grothendieck topology on C containing $T_1$ and $T_2$). ...
3
votes
0answers
512 views

Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...
5
votes
4answers
453 views

Can isomorphisms of schemes be constructed on formal neighborhoods?

Let (A,m) be a complete local Noetherian ring and let X and Y be two schemes of finite type over A (and flat over A). Let Xn and Yn be the reductions of X and Y mod mn+1. Question: Suppose there ...
2
votes
1answer
296 views

Different ways to construct maps and the tensor products of line bundles

Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective ...
1
vote
0answers
277 views

Finding a metric on a tubular neighborhood of an embedded surface

Hey all. The setup for my question is an embedded surface $\Sigma \hookrightarrow M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_{\Sigma}$ on $\Sigma$, I would like ...
41
votes
0answers
1k views

To what extent does Spec R determine Spec of the Witt vector ring over R?

Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors -- PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from ...
7
votes
2answers
220 views

What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)

Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
0
votes
0answers
29 views

Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by ...
0
votes
0answers
19 views

Splitting of totally geodesic Riemannian foliations

Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...
3
votes
1answer
70 views

geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$

Let $G$ be Lie group and $K \subset G$ a closed subgroup, such that there exists a $v \in T(G/K)$ whose isotropy-group $G_v$ is discrete (so iff $\dim G_v =0$). Lets assume $g$ acts properly on ...
5
votes
1answer
162 views

Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations. Theorem The normal subgroups of $S_\infty$ are ...
7
votes
1answer
115 views

What are types of coalgebras that are more naturally described by cooperads?

Some background. Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of ...
4
votes
1answer
272 views

Normalization of complete intersection

Let $A$ be an integral complete local ring over a field which is complete intersection. Let $B$ be a normalization of $A$. Q. Is $B$ Gorenstein? I guess that even the normalization of Gorenstein ...
0
votes
1answer
73 views

Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item. In this setting it is relevant what is the distribution of the values of the ...
9
votes
2answers
277 views

Homotopy groups of Moore spaces

Is there anything known about the homotopy groups of the Moore spaces $M(\mathbb Z_m,n)$ if $m\neq 2$ and $n \geq 2$?
3
votes
0answers
49 views

Cubic Directed Cayley graphs of 2-generated torsion-free groups

Is there a torsion-free group $G=\langle x,y \rangle$ such that the directed Cayley graph $\Gamma=Cay(G,\{x,y,x^{-1}y\})$ contains a finite cubic induced subgraph? The vertex set of $\Gamma$ is $G$ ...
7
votes
1answer
112 views

A clarification on pointed Gromov-Hausdorff convergence

According to Burago-Burago-Ivanov, one says that the sequence of pointed metric spaces $(X_n,d_n,p_n)$ GH-converges to $(X,d,p)$ if for every $R>0,\varepsilon>0$, there exists a $N$ such that ...
14
votes
1answer
245 views

Does the injection $\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ split?

Let $F_n$ be the free group on $n$ letters. The question is as in the title: letting $i:\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ be the natural injection, does there exist a homomorphism ...
10
votes
3answers
558 views

The universe of sets, existential quantification in set theory

Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context. In ZF one can prove $\not\exists x (\forall y (y\in x)).$ ...
50
votes
1answer
2k views

Did Bourbaki write a text on algebraic geometry?

Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?
26
votes
65answers
9k views

Fiction books about mathematicians? [closed]

What are some fiction books about mathematicians? It seems to me rather difficult for writers to create good books on this subject. Some years ago I thought there were no such books at all. There ...
57
votes
16answers
11k views

f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential

The question is about the function f(x) so that f(f(x))=exp (x)-1. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here ...
25
votes
3answers
2k views

Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ... On the other hand model theory, in particular after Hrushovski, found many ...
68
votes
8answers
12k views

Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ ...
73
votes
4answers
12k views

Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.) Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...
73
votes
4answers
9k views

How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
13
votes
1answer
609 views

Homology theory represented by Madsen-Tillman spectra

The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of ...
35
votes
2answers
4k views

“Closed-form” functions with half-exponential growth

Let's call a function f:N→N half-exponential if there exist constants 1<c<d such that for all sufficiently large n, cn < f(f(n)) < dn. Then my question is this: can we prove that no ...
14
votes
3answers
1k views

How to prove that every real number is a zero of some power series with rational coefficients (if true)

How would one approach proving that every real number is a zero of some power series with rational coefficients? I suspect that it is true, but there may exist some zero of a non-analytic function ...
0
votes
1answer
696 views

When may Function (meromorphic) be expanded as power series with coefficients of integers

Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form: $$F=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M ...
10
votes
2answers
470 views

Cohen-Macaulay domain with non-Cohen-Macaulay normalization?

Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...
0
votes
0answers
172 views

Existence of flat models of a smooth finite type algebra over $R((t))$

Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary), $B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth. Up to this generality, can one ...
2
votes
1answer
133 views

normalization of a bijection

Let $X^\nu,Y^\nu$ be normalizations of affine varieties $X$ and $Y.$ If a morphism $f:X\to Y$ is a bijection, does it imply that its lift $f^\nu: X^\nu\to Y^\nu$ is an isomorphism?
3
votes
1answer
346 views

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
13
votes
1answer
460 views

How can I tell if a variety is normal?

Suppose $R$ is a subalgebra of ${\mathbb C}[x_1,...,x_N]$ generated by polynomials $p_1,...,p_k.$ I know that ${\mathbb C}[x_1,...,x_N]$ is the integral closure of $R$. Is there an algorithm to ...
7
votes
2answers
517 views

Cayley Transform for all reductive groups a.k.a an algebraic logarithm

Is it true that for every reductive algebraic $G$ over ${\mathbb C}$ with a Lie algebra $\mathfrak g$ there is an open neighborhood $U$ of the identity in $G$ and an algebraic function (in a sense of ...
5
votes
1answer
312 views

What is known about maximal free subgroups of surface groups?

Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...
36
votes
1answer
9k views

Lecture notes by Thurston on tiling

I am looking for a copy of the following W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes. I see that a lot of papers in the tiling literature refer to it but I ...

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