2
votes
1answer
77 views

On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category. Question: What about the converse, i.e., can we characterize every unitary modular tensor ...
0
votes
0answers
48 views

Formulation of constraints

I would like to relax the following binary SDP problem $$ \max_{A,x_i,y_i} \ trace(A) $$ $\quad \qquad \qquad $ subject to: $$ A + \sum_{i=1}^N x_i D_i + \beta \sum_{i=1}^M y_i D_i \preceq \alpha .I ...
0
votes
0answers
4 views

For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
0
votes
1answer
101 views

Right inverse of the Seiberg-Witten functional

For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e. $$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...
0
votes
1answer
82 views

An inequality of real continuous function with f'>0 and f''>0

I proposed my conjecture as follows: Let $f(x)$ is a real continuous function on $[m, M]$ and $f'>0, f''>0$ on $[m, M]$, let $m \le x_i \le M$, for $i=1, 2,..., n$. Then $$\frac{f(x_1)+f(x_2)+...
2
votes
0answers
127 views

What techniques are available for constructing D-modules over smooth projective varieties?

I'm trying to learn about D-modules for computing intersection cohomology but I'm having trouble coming up with explicit constructions of D-modules on projective varieties. Since this is an involved ...
2
votes
0answers
22 views

Covering Number of a Positive Semidefinite Cone (Approximate the Objective of a SDP)

I was wondering what the covering number of a positive semidefinite cone is. Consider the semidefinite optimization program \begin{align} \max\langle \mathbf{C}, \mathbf{X} \rangle~~\text{subject to}~...
3
votes
0answers
126 views

Research topics in Curves and Surfaces [on hold]

I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry ...
6
votes
0answers
132 views

Where can I find basic “computations” of equivariant stable homotopy groups?

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (...
3
votes
2answers
142 views

Combinatorial identity involving number of cycles (of any length) in a permutation

I am going through Phil Hanlon's paper and on page 127, right after the first paragraph, "It is well known that.." which boils down to the following identity: $$ \prod_{i=0}^{n-1}(\beta-i) = \sum_{\...
1
vote
0answers
50 views

About Composition diamond lemma

Composition Diamond lemma for Lie algebra over a field $F$ has already been investigated in several papers : L.Bokut and Y.Q.Chen Groebner-Shirshov bases for Lie algebras and A.I Shirshov, ...
4
votes
3answers
268 views

($\oplus$, $\otimes$) is a semiring. If $\otimes$ = +, what are the possible operators $\oplus$?

Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals. If $\otimes$ is +, what are the possible operators for $\oplus$? So far I have proven that ...
0
votes
0answers
60 views

Reference request for a well-known lemma in Parabolic Vector Bundle

In the paper- "Moduli Space of parabolic vector bundles on a curve" - Usha N Bhosle, Indranil Biswas-Beitr Algebra Geom (2012), 53:437-449, DOI: 10.1007/s13366-011-0053-7, Lemma $2.1$ is being ...
3
votes
1answer
136 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
0
votes
1answer
70 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $0 \le a_{j,j} \le 1$ and $-1 \le ...
0
votes
0answers
32 views

Moduli space of Parabolic Vector Bundles with arbitrary parabolic weights

I have just posted another question related to Moduli Space of Parabolic Vector Bundles on Curve. The questions came up when I was trying to read the paper (Desingularisation of the Moduli Varieties ...
1
vote
1answer
70 views

Isofibrations and Diagonal Functors

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor. Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any ...
-3
votes
0answers
53 views

Fredholm operators: how to calculate Coker and Ker [on hold]

Exercise: Let $1\leq p \leq \infty$. For each $n\in\mathbb{Z}$ construct a Fredholm operator $F:l^p\to l^p$ whose index is $n$. Solution (given in the lecture classe): $F_n(x_i):=(0,\ldots, 0,x_1,...
-1
votes
0answers
40 views

Direct product between joins of subgroups [on hold]

Suppose that $G$ is a finite group with $A, B, H, K \leq G$. Suppose that $H\times A \leq G$ and $K\times B \leq G$. I want to show that $\langle H,K \rangle \times \langle A, B \rangle = \langle H \...
3
votes
1answer
94 views

Linear independency and compactness of the set of pure states of a $C^*$-algebra

Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states. Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...
5
votes
0answers
138 views

An inequality in cyclic polygon and tangential polygon

I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following: Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let P be ...
3
votes
0answers
64 views

Extending homomorphisms between ordered abelian groups

Let $\Omega$ be a linearly (i.e. fully) ordered set, and let $\Lambda_{\Omega}$ be the ordered abelian group consisting of those $(\lambda_\omega)_{\omega\in\Omega}\in\mathbb{R}^{\Omega}$ with well-...
11
votes
1answer
449 views

Is a one-dimensional compact complex analytic space necessarily projective?

Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is ...
6
votes
1answer
149 views

Group bundles for topological spaces without universal cover

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first describe the situation: There are two approaches in defining Homology with local coefficients of a ...
0
votes
1answer
116 views

Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold. Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
6
votes
0answers
222 views

Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g. A question about ordinal definable real numbers . A negative answer was obtained in Archive for ...
4
votes
1answer
231 views

A generalization of Erdős–Mordell inequality [on hold]

I proposed my conjecture generalization of Erdős–Mordell inequality as following: Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ ...
-1
votes
0answers
24 views

Hyperbola application [on hold]

A curved mirror is placed in a store for a wide angle view of the room. the right hand branch of x squared over one minus y squared over three equals one models the curvature of the mirror. a small ...
0
votes
0answers
20 views

Algorithm for Longest Common Subtour

For a new kind of heuristic for the TSP I need to calculate the longest subtour, that is common to a set $T_1,\ ...,\ T_m$ of tours, that are "good" approximations of the optimal tour $T_{opt}$. By a ...
-1
votes
0answers
54 views

How many number of finite points exists inside the circle? [on hold]

I am doing project on Image processing dealing with circular images. So I need an approximate number of pixels present inside circle image of radius R and Circle center of (x,y). Please give me the ...
0
votes
1answer
63 views

Computing canonical forms from orbit partitions

Suppose we know the orbit partition of the vertices of a graph (due to the action of its automorphism group). Is it easy (as in "polynomial time") to generate a canonical form (aka "canonical labeling"...
1
vote
1answer
83 views

Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$. Does it always follow that $$...
0
votes
0answers
91 views

Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold. If it is necessary, we could consider the $b_1(Y)=0$ case. Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
3
votes
0answers
62 views

Intersection multiplicity of limit linear spaces

Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$. Now let $\Lambda_{\xi, q}$, with $\xi \...
6
votes
1answer
247 views

The current status of the conjecture on algebraic matroids

Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic? Thank you!
0
votes
0answers
25 views

On balanced bipartite graphs

Of the $2^{n^2}$ balanced bipartite graphs on $2n$ vertices how many of them have $i$ perfect matchings where $i\in\Bbb N\cup\{0\}$ and $0\leq i\leq n!$ holds?
1
vote
0answers
44 views

A priori estimates for elliptic operators

Suppose $L : L^{m,p}(M)\rightarrow L^p(M)$ is some elliptic operator of order $m$, and $(M,g)$ is a compact Riemannian manifold. Then it is known that there exists a constant $C$ such that we have the ...
21
votes
0answers
621 views

Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...
0
votes
0answers
88 views

Sum of unit vectors always has a binary span after constrained permutations

Conjecture: Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$. An enumeration $ E \cup -E = \{f_1, \ldots, ...
8
votes
0answers
196 views

A Combinatorial Game: the Snake and the Hunter

The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...
0
votes
0answers
99 views

Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal. This is one of the many equivalent formulations of the Birch and ...
3
votes
1answer
87 views

Isoperimetric inequality via Crofton's formula

I have seen various assertions that one can derive the isoperimetric inequality in the plane from Crofton's formula in geometric probability. Unfortunately, I have not managed to figure out such a ...
6
votes
1answer
208 views

The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$

Assume that $M$ is an arbitrary manifold. Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one? If not, what is a counter ...
-1
votes
1answer
135 views

number of partitions from 0 to n^2 [on hold]

You are given the numbers 0 to n^2. You must use n numbers with no number greater than n to form all the partitions of the numbers 0 to n^2. For example with n=4 you want to find the partitions of 7:...
-3
votes
1answer
152 views

Does one need an external, peer-reviewed grant to become tenured faculty in this field? [on hold]

As a secondary question, how important is it to be awarded grants to remain employed as a PhD mathematician at an academic research institution? Is it common or uncommon for PhD mathematicians to have ...
2
votes
0answers
77 views

Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is ample. then there is a Kahler metric $\omega_E$ such that $Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then $h=\frac{1}{\...
3
votes
1answer
121 views

Elements in the group von Neumann algebra which are not summable

Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \...
0
votes
1answer
50 views

Count Functional digraph [on hold]

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...
1
vote
0answers
46 views

Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that $$S^°=\{s\in S| f\text{ is ...
8
votes
2answers
260 views

How to compute $[CP^2, G/PL]$?

Let $E^4$ be the two stage Postnikov space appearing in the homotopy type of the classifying space $G/PL$. One of its properties is that it only has two nontrivial homotopy groups $\pi_2(E)=Z/2Z$ and $...

15 30 50 per page