Related papers: A filter on a collection of finite sets and Eberle…
We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class…
Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $\omega_1$ must be somewhere $\omega_1$-dense. We prove a dichotomy…
Given a graph $G$, we define a filtration of simplicial complexes associated to $G$, $\mathcal{F}_0(G)\subseteq\mathcal{F}_1(G)\subseteq\cdots\subseteq\mathcal{F}_\infty(G)$ where the first complex is the independence complex and the last…
A set of natural numbers will be called \emph{harmonic} if the reciprocals of its elements form a divergent series. An ultrafilter of the natural numbers will he called \emph{harmonic} if all each members are harmonic sets. The harmonic…
For each closed, positive (1,1)-current \omega on a complex manifold X and each \omega-upper semicontinuous function \phi on X we associate a disc functional and prove that its envelope is equal to the supremum of all…
Given a finite subset $\Sigma\subset\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma;q)=\big\{\sum_{n=0}^\infty…
We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of…
We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their…
We study Baire category for subsets of 2^omega that are downward-closed with respect to the almost-inclusion ordering (on the power set of the natural numbers, identified with 2^omega). We show that it behaves better in this context than…
Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space of tilings of M, with finite local complexity (relative to some symmetry group Gamma) and closed in the natural topology. Then Omega is the inverse limit of a…
Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$. We prove an integral representation for local functionals with respect to $X$ and a result…
Given positive integers $e_1,e_2$, let $X_i$ denote the set of $e_i$-dimensional subspaces of a fixed finite vector space $V=(\mathbb{F}_q)^{e_1+e_2}$. Let $Y_i$ be a non-empty subset of $X_i$ and let $\alpha_i=|Y_i|/|X_i|$. We give a…
We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we…
We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective…
We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a…
We strengthen, in various directions, the theorem of Garnett that every $\sigma$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen…
We prove that if $A$ is a singular MASA in a II$_1$ factor $M$ and $\omega$ is a free ultrafilter, then for any $x\in M\ominus A$, with $\|x\|\leq 1$, and any $n\geq 2$, there exists a partition of $1$ with projections $p_1, p_2, ...,…
We shall establish some properties of thin $\sigma$-ideals of compact sets in compact metric spaces (in particular, the $\sigma$-ideals of compact null-sets for thin subadditive capacities), and we shall refine the celebrated theorem of…
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$.…
We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted…