Related papers: Hedetniemi's conjecture from the topological viewp…

In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then…

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the…

In this note, we derive a uniqueness theorem for minimal graphs of general codimension under certain restrictions closed related to the convexity (not strict convexity) of the area functional with respect to singular values, improving the…

Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal index of A: $I(A) = \max \{\lambda \ge 0 : A\circ B \ge \lambda B$ for all $0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} = A_{ij}…

A short proof of the Caratheodory conjecture about index of an isolated umbilic on the convex 2-dimensional sphere is suggested. The argument is based on the study of geodesic lines near cone-type singularity of a metric induced by…

In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham…

Box complex is a $\mathbb{Z}_2$-space associated to a graph, and it is known that a certain $\mathbb{Z}_2$-homotopy invariant of it, called the $\mathbb{Z}_2$-index, gives an effective lower bound for the chromatic number. On the other…

We prove that the complement of any affine 2-arrangement in R^d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane…

Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang's version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka…

The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two…

We produce a criterion for open sets in projective $n$-space over a separably closed field to have \'etale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which \'etale cohomological dimension is…

We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is…

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split…

Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous.…

Let $\gamma(S_n)$ be the minimum number of proper subgroups $H_i$ of the symmetric group $S_n$ such that each element in $S_n$ lies in some conjugate of one of the $H_i.$ In this paper we conjecture that…

For given positive integers $d$ and $m$, consider the projective klt pairs $(X,B)$ of dimension $d$, of Cartier index $m$, and with semi-ample $K_X+B$ defining a contraction $\pi\colon X\to Z$. We prove that it is not possible in general to…

In 1973 Montgomery formulated the pair correlation conjecture, predicting that the local spacing statistics of the nontrivial zeros of the Riemann zeta function coincide with those of eigenvalues of large Hermitian matrices from the…

In the first part of this paper, we give a new look at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones' index. The minimal conditional expectation is characterized by means of a canonical state on the…

A subbundle of a Hermitian vector bundle $(E, h)$ can be metrically and differentiably defined by the orthogonal projection onto this subbundle. A weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by definition, an…

For a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K\supset \mathbb C$, we deduce from the compatibility between cup products Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an alternative way…