Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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3 votes
1 answer
765 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
2 votes
0 answers
273 views

An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
5 votes
3 answers
686 views

Reference request about the representations of the group $PSL_2(\mathbb{F}_q)$

Is there a review/exposition of the representation theory of $PSL_2(\mathbb{F}_q)$ ? Like an enumeration of its irreducible representations and their dimensions as a function of $q$?
4 votes
1 answer
1k views

Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $...
0 votes
0 answers
81 views

Good covering of a (singular) curve in a complex surface

Let $W$ be a $2$-dimensional complex manifold and $C\subset W$ a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection $\{V_j\}...
7 votes
0 answers
358 views

Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...
0 votes
1 answer
578 views

Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings: Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation $$ B\...
6 votes
1 answer
651 views

Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a ...
2 votes
1 answer
108 views

Renorming into contraction

In Pazy's book on semigroups he mentions (page 18) that when you have a commuting family of operators $B(t)$, such that $$ \sup \| B(t_1) .. B(t_n) \| \le M $$ for all finite choices $t_1, .. t_n$ ...
2 votes
1 answer
335 views

Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?

This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce ...
2 votes
1 answer
585 views

Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $\mathcal{A}$, $\mathcal{B}$ be two triangulated categories and $F: \mathcal{A}\to \mathcal{B}$ be a triangulated functor between them. Then $F$ induces an homomorphism between their Grothendieck ...
2 votes
1 answer
343 views

Generators of the colored braid group (two colors), reference

I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white. It is easy to find a set of generators for $B_{n,n}$: $$ \begin{cases} \...
3 votes
0 answers
280 views

Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
20 votes
1 answer
2k views

When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
4 votes
1 answer
297 views

Young tableau with no i in row i, name that derangement

This question has its genesis in a group assignment: $k$ students are to be given oral exams. Each student will be asked one distinct question from $n$ questions given to them earlier, no two students ...
3 votes
0 answers
131 views

Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form $$\ddot{...
2 votes
0 answers
243 views

Simply connected Kahler manifold without any effective divisor

Does anyone know an example of a simply-connected compact Kahler manifold without an effective divisor? Does anyone know a reference on this topic? Thanks!
1 vote
1 answer
1k views

Sum of two unbounded self-adjoint operators

Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are ...
4 votes
0 answers
122 views

About some distributive laws in the Bousfield lattice

It is know that for any $\alpha$-well generated tensor triangulated category $\mathcal{T}$ the collection of Bousfield classes forms a set, furthermore this set is a complete lattice, denoted by $\...
7 votes
1 answer
268 views

On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer? Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\{...
2 votes
0 answers
153 views

Size of the eigenfunction of Laplacian (reference request)

It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then $$||\phi||_{L^\...
5 votes
2 answers
293 views

Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite

I am trying to understand the decomposition $$L^2(SL_2(\mathbb{C})/\Gamma)=\oplus_i C_i \oplus V_{temp}$$ where $C_i$ are complementary series representations corresponding to eigenfunctions of the ...
10 votes
1 answer
1k views

Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...
5 votes
2 answers
843 views

Is every positive integer a sum of at most 4 distinct quarter-squares?

There appears to be no mention in OEIS: Quarter-squares, A002620. Can someone give a proof or reference? Examples: quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$ 2-term sums: ${2+1, 4+1, ...
11 votes
2 answers
2k views

Central extension of the algebraic loop group

I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but ...
4 votes
1 answer
517 views

Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
1 vote
0 answers
61 views

Reference for using an algebra of meromorphic functions to extend a Lie algebra

For example, let $\mathfrak{g}=\mathfrak{sl}_{2}\left(\mathbb{C}\right)$, let $s_{0}=1$, $s_{1}=-1$, $s_{2}$=0, $s_{3}=\infty$ in $\mathbb{P}_{1}\left(\mathbb{C}\right)$ and $\mathcal{R}$ is the ...
4 votes
0 answers
121 views

Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, u\...
16 votes
1 answer
1k views

Has this strong number theoretic conjecture of Euler been proved, and where could I find such a proof?

Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration. It is on pages 90-98 of "Induction and Analogy in ...
27 votes
1 answer
1k views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of length $<\...
6 votes
1 answer
305 views

Tunnel number of Pretzel knots

I would like to know the tunnel number of $n$-pretzel knots. I have searched and found nothing for any $n>3$. When $n=2$, $t(K)=1$ or $2$ depending on the number of twists, which is proved in a ...
-4 votes
1 answer
572 views

What's the minimum amount of knowledge to start doing research? [closed]

There are cases in which you have too much knowledge of something to do anything interesting ,and cases in which a lack of experience with a problem (and the prejudices about it) helps someone solve ...
3 votes
1 answer
159 views

Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...
6 votes
2 answers
460 views

Centralizers of reflections in special subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with $m_{i,j}=m_{j,i}$...
1 vote
0 answers
194 views

Seeking reference to result in this talk by Voevodsky [duplicate]

In this presentation by Vladimir Voevodsky [1], he mentions a result that there is a formula over the natural numbers with a single free variable such that one can prove that there is no algorithmic ...
2 votes
0 answers
163 views

Derived categories of modules categories

Does anyone know if there is a note or a paper about the derived category of the category $\sigma[M]$ where $M$ is a left module over a ring?, and some uses of this.
4 votes
0 answers
213 views

Fixed Points of the Friedman Stanley Jump

Consider the situation of a pair $(X,E)$, where $X$ is a standard Borel space and $E$ is an invariant equivalence relation on $X$*. The Friedman-Stanley jump of this pair is an equivalence relation $...
7 votes
2 answers
524 views

Constructing Ramanujan graphs from elliptic curves

Is there an exposition which explains how to do this step-by-step? (I see stray references and allusions to such a thing being possible but can't locate anything concretely) Something to do with ``...
27 votes
3 answers
3k views

Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
14 votes
2 answers
2k views

What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?

Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several ...
1 vote
0 answers
167 views

Estimates of entropy of functional spaces

Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it. ...
8 votes
1 answer
603 views

Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following. There exist $\Sigma$-free operads $\mathcal{C}...
9 votes
1 answer
634 views

What happens to continuous spectrum upon discretization?

Excuse me for a bit of an vague question, but I haven't been able to find a definite answer for this for quite some time. My question is regarding (mostly non-normal )linear operators and their ...
18 votes
2 answers
920 views

Vanishing of Dolbeault cohomologies and Steinness

That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
4 votes
0 answers
294 views

Is there some absoluteness between the Boolean valued universe $V^{B}$ and $V$?

It is well known that if $\phi$ is a $\Delta_{1}$-formula and $x_{1},..,x_{n}$ in $V$ and $V[G]$ is a forcing extension, then $V\models\phi(x_{1},...,x_{n})$ if and only if $V[G]\models\phi(x_{1},...,...
4 votes
1 answer
256 views

Reference request for division algebras, over $\mathbb{Q}_{p}((t))$

I was looking for a possible reference that would answer the following question, Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over $\...
3 votes
0 answers
431 views

Kahler identities on almost Kahler manifolds

Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to \Omega_{X}^{p,q+...
3 votes
0 answers
345 views

How does one compute a colimit of monoidal categories?

The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories. Here's a guess: In order to compute a colimit of monoids we can push everything down ...
1 vote
0 answers
98 views

Name for condition on map of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that $k(\epsilon)=\epsilon$ for all $a,b\in M$, there exists $v\in N$ such that ...
10 votes
2 answers
924 views

Morgan Shalen compactification of $\mathbb C^2$

I'm reading the Otal's survey on the compactification of Morgan Shalen. (available here) He claims in an example (page 8) that the compactification of $\mathbb C^2$ is $S^4$, which sounds completely ...

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