Related papers: Sarnak's Conjecture for nilsequences on arbitrary …

Let $X_N$ be a random trigonometric polynomial of degree $N$ with iid coefficients and let $Z_N(I)$ denote the (random) number of its zeros lying in the compact interval $I\subset\mathbb{R}$. Recently, a number of important advances were…

Let $\mathbb{T}^\omega$ be the infinite-dimensional torus, and $T: \mathbb{T}^\omega\to \mathbb{T}^\omega$ be defined by \[ T: (x_1, x_2, \dots, x_k, \ldots) \mapsto (x_1 + \alpha, x_2 + h(x_1), \dots, x_k + h(x_1 + (k-2)\beta), \dots) \]…

If $G$ is a finite group, the Grothendieck group ${\mathbf{K}}\_G(G)$ of the category of $G$-equivariant ${\mathbb{C}}$-vector bundles on $G$ (for the action of $G$ on itself by conjugation) is endowed with a structure of (commutative)…

W. T. Gowers proved that every Lipschitz function from the unit sphere of the Banach space $c_0$ to $\mathbb{R}$ is oscilation stable. His proof uses a result about finite partitions of the set $FIN_k$ of finitely supported functions $p$…

In this paper we introduce an algebra embedding $\iota:K< X >\to S$ from the free associative algebra $K< X >$ generated by a finite or countable set $X$ into the skew monoid ring $S = P * \Sigma$ defined by the commutative polynomial ring…

In this article we use existing machinery to define connective $K$-theory spectra associated to topological ringoids. Algebraic $K$-theory of discrete ringoids, and the analytic $K$-theory of Banach categories are obtained as special cases.…

The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$ has degree…

In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete K\"ahler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of…

We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra…

A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$.…

The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an…

We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case…

Let $\mathbb K$ be a field of characteristic zero. We prove that its motivic cohomology in degree $m-1$ and weight $m$ is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin…

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring…

We generalize a formula due to Macdonald that relates the singular Betti numbers of $X^{n}/G$ to those of $X$, where $X$ is a compact manifold and $G$ is any subgroup of the symmetric group $S_{n}$ acting on $X^{n}$ by permuting…

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions…

Let $K$ be a field and $q\in K^{\times}$. Let $e$ be the multiplicative order of $q$; or 0 if $q$ is not a root of unity. Let $\bQ:=(q^{v_1},...,q^{v_r})$. Let ${K}_r(n)$ be the set of Kleshchev $r$-multipartitions with respect to…

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of…

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved,…

We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of Andr\'e's generalization of the Grothendieck period conjecture, which we…