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Let $\Gamma$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $\rho:\Gamma\to G_2$ and a surprising role played by higher rank…

We characterize the model spaces $K_\Theta$ in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of…

For quantum field theories with topological sectors, Monte Carlo simulations on fine lattices tend to be obstructed by an extremely long auto-correlation time with respect to the topological charge. Then reliable numerical measurements are…

Given any function $\phi \colon [0,\infty)\to (0,1]$ satisfying $\lim_{\xi\to\infty}\phi(\xi) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose…

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset…

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant…

Let $[q] = \{0,1,\ldots,q-1\}$, let $\Delta[q]$ denote the simplex of probability measures on $[q]$, and let $\gamma$ denote the Lebesgue measure normalized on $\Delta[q]$. We prove that for any symmetric monotone function $f \colon[q]^n…

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of…

We give two characterizations, one for the class of generalized Young measures generated by $\mathcal A$-free measures, and one for the class generated by $\mathcal B$-gradient measures $\mathcal Bu$. Here, $\mathcal A$ and $\mathcal B$ are…

We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\FI}) standard probability measure…

Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free…

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather…

We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables,…

In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature. However, the notion of curvature is often pessimistic, and we rarely…

We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo's result for Wigner matrices having the same type of entries [7]. To…

Suppose that $S_1$ and $S_2$ are nonempty subsets of a complete metric space $(\mathcal{M},d)$ and $\phi,\psi:S_1\to S_2$ are mappings. The aim of this work is to investigate some conditions on $\phi$ and $\psi$ such that the two functions,…

This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\ni\gamma$-torsional rigidity $\mathcal{T}_{\gamma,\mathsf{g}}$ on a complete Riemannian…

We discuss Meyers-Serrin's type results for smooth approximations of functions $b=b(t,x):\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^m$, with convergence of an energy of the form \[ \int_{\mathbb{R}}\int_{\mathbb{R}^n} w(t,x)…

We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly…

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of \(T^{\rtimes}\)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary…