Related papers: Carlsson's rank conjecture and a conjecture on squ…

Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is…

For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a…

We give a short proof that Strassen's asymptotic rank conjecture implies that for every $\varepsilon > 0$ there exists a $(3/2^{2/3} + \varepsilon)^n$-time algorithm for set cover on a universe of size $n$ with sets of bounded size. This…

It is shown that the rank-$k$ numerical range of every $n$-by-$n$ complex matrix is non-empty if $n \ge 3k - 2$. The proof is based on a recent characterization of the rank-$k$ numerical range by Li and Sze, the Helly's theorem on compact…

A function from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ is $k$th order sum-free if the sum of its values over each $k$-dimensional $\mathbb{F}_2$-affine subspace is nonzero. It is conjectured that for $n$ odd and prime,…

We consider certain q-series depending on parameters (A,B,C), where A is a positive definite r times r matrix, B is an r-vector and C is a scalar, and ask when these q-series are modular forms. Werner Nahm conjectured a criterion for which…

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We…

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR compactum and x\in X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of…

Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of rank r on C. For each integer r'<r, associate to E the invariant s_{r'}(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal degree. For every…

We develop a new method to deal with the Cancellation Conjecture of Zariski in different environments. We prove the conjecture for free associative algebras of rank two. We also produce a new proof of the conjecture for polynomial algebras…

Let $K$ be any field and $x = (x_1,x_2,\ldots,x_n)$. We classify all matrices $M \in {\rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${\rm rk} M \le 2$. As a special case, we describe all such matrices…

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that…

Quadratic conjecture is a strengthening of oliver's $p$-group conjecture. Let $G$ be a $p$-group of maximal class of order $p^n$. We prove that if $n\le 8$ or $n\ge \max\{2p-6,p+2\}$ then $G$ satisfies Quadratic Conjecture. Hence quadratic…

We give an upper bound for the rank $r$ of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if $r=2^a\cdot b$ with $b$ odd, then $r\le 9$ for $a=0$, $r\le 10$…

Let K be an infinite field such that its characteristic is not 2. We show that, for every $A\in\mathcal{M}_n(K)$ such that $\mathrm{rank}(A)\geq n/2$, there exists $B\in\mathcal{M}_n(K)$ such that $B$ is similar to $A$ and $A+B$ is…

Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms between open subsets of $N$, and more generally…

There is a long-standing conjecture which states that every uniform algebraic vector bundle of rank $r<2n$ on the $n$-dimensional projective space $\mathbb{P}^n$ over an algebraically closed field of characteristic $0$ is homogeneous. This…

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k…