Related papers: On spaces with a $\pi$-base whose elements have an…

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in ${\mathbb P}^{M}$, where $M\geq 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the parameter…

An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero…

A hyperk\"ahler manifold is defined as a Riemannian manifold endowed with three covariantly constant complex structures that are quaternionically related. A twistor space is characterized as a holomorphic fiber bundle $p: \mathcal{Z}…

We prove the following result: For each closed $n$-dimensional manifold $M$ in a (finite or infinite-dimensional) Banach space $B$, and each positive real $m\leq n$ there exists a pseudomanifold $W^{n+1}\subset B$ such that $\partial…

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection…

We study closure properties for the Littlestone and threshold dimensions of binary hypothesis classes. Given classes $\mathcal{H}_1, \ldots, \mathcal{H}_k$ of Boolean functions with bounded Littlestone (respectively, threshold) dimension,…

We study a Poisson structure $\pi$ on the Grothendieck resolution $X$ of a complex semi-simple group $G$ and prove that the desingularization map $\mu:(X,\pi) \to (G,\pi_0)$ is Poisson, where $\pi_0$ is a Poisson structure such that…

We say that a topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open sets $\{U_n\mid n\in\omega \}$ of $X$, we can find $x_n\in U_n$ such that the sequence $(x_n)$ has no convergent subsequences.…

Refining an earlier result due to Hahlomaa, we provide a new Carleson-type condition for $k$-regular sets in the Heisenberg group $\mathbb{H}^n$ to have big pieces of Lipschitz images of subsets of $\mathbb{R}^k$ for $1\leq k\leq n$. Our…

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such…

For a connected, locally path connected space $X$, let $H$ be a subgroup of the fundamental group of $X$, $\pi_1(X,x)$. We show that there exists an open cover $\cal U$ of $X$ such that $H$ contains the Spanier group $\pi({\U},x)$ if and…

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0<s\leq1$, $1<p<\infty$ with $sp>2$ and a Lipschitz domain $\Omega\subset \mathbb{C}$, the…

The aim of this paper is to study the Frechet-Urysohn property of the space $Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a Hausdorff space $X$, endowed with the pointwise convergence topology. It is proved that…

We prove that if $\Phi:X\to Y$ a mapping of weak bounded length distortion from a quasiconvex and complete metric space $X$ to any metric space $Y$, then for any Lipschitz mapping $f:\mathbb{R}^k\supset E\to X$ we have that ${\mathcal…

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However,…

On a compact Riemann surface $\Sigma$ of genus $g>1$, equipped with a complex vector bundle $E$ of rank $2$ and degree zero let $M_H$ be the moduli space of Higgs bundles. $M_H$ admits a $\mathbb C^{\star}$-action and to each stable…

Let $X$ be a Calabi--Yau manifold and $Q\subset X$ a closed connected embedded special Lagrangian; closed Lagrangians mean compact Lagrangian submanifolds without boundary. We prove that if the fundamental group $\pi_1Q$ is abelian then…

We introduce a notion of the Bergman-Shilov (or Shilov) boundary for some subclasses of upper-semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of…

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$…

Let $E$ be a complete Hausdorff locally convex space over $\mathbb{C}_{p},$ let $A\in\mathcal{L}(E)$ such that $(I-\lambda A)^{-1}$ is analytic on its domain. In this paper, we give a necessary and sufficient condition on the resolvent of…