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Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of…

We initiate a classification of uniform measures in the first Heisenberg group $\mathbb H$ equipped with the Kor\'anyi metric $d_H$, that represents the first example of a noncommutative stratified group equipped with a homogeneous…

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to…

Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by…

Functions, uniformly bounded in $BV$ norm in some bounded open set $U$ in $R^n$, are compact in $L_1(U)$. This result is known when $U$ has Lipschitz boundary [EG Th. 4 p. 176], [G 1.19 Th. p. 17], [Z 5.34 Cor. p. 227]; the proof for…

Let $M$ be a compact $d$-dimensional Riemannian manifold without a boundary. Given $E \subset M$, let $\Delta_{\rho}(E)=\{\rho(x,y): x,y \in E \}$, where $\rho$ is the Riemannian metric on $M$. Let $\Delta_{\rho}^x$ denote the pinned…

Let $f$ be a symmetric norm on ${\mathbb R}^n$ and let ${\mathcal B}({\mathcal H})$ be the set of all bounded linear operators on a Hilbert space ${\mathcal H}$ of dimension at least $n$. Define a norm on ${\mathcal B}({\mathcal H})$ by…

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the…

To every closed subset $X$ of a symplectic manifold $(M,\omega)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,\omega)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,\omega}$, generalizing the Hofer norm. We…

Gr\"unbaum's inequality guarantees that the centroid of a convex body has halfspace depth at least $1/e$: every halfspace containing the centroid captures at least a $1/e$ fraction of the body's volume. For mixed-integer convex sets…

We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central…

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the…

We consider collections of Lagrangian submanifolds of a given symplectic manifold which respect uniform bounds of curvature type coming from an auxiliary Riemannian metric. We prove that, for a large class of metrics on these collections,…

We introduce the concept of ODD ('$\mathbf{O}$rthogonally $\mathbf{D}$egenerating on a $\mathbf{D}$ivisor') Riemannian metrics on real analytic manifolds $M$. These semipositive symmetric $2$-tensors may degenerate on a finite collection of…

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the…

Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space.…

Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces:…

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set…

We study an $\mathcal{N}=2$, pure $U(1)$ SYM theory on a Gibbons-Hawking space $\Omega$-deformed using the $U(1)$ isometry. The resultant 3D theory, after an appropriate "Nekrasov-Witten" change of variables, is asymptotically equivalent to…