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We consider an $n$ by $n$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix. We prove that for $k\sim n^{1/6}$ and $b^2:=\frac{1}{n}\operatorname{Tr}(|T|^2)$, as $n$…
Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $\sigma_2$-curvature, normalized by the volume, is uniquely (up to M\"obius transformations) minimized by…
In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansion. This estimate is an analog of a result previously obtained by…
For any 3-manifold M and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form S^3/Gamma we…
We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand…
The optimization of shape functionals under convexity, diameter or constant width constraints shows numerical challenges. The support function can be used in order to approximate solutions to such problems by finite dimensional optimization…
A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all…
We study invariant measures of continuous contact model in small dimensional spaces ($d =1,2$). Under general conditions we prove that in the critical regime this system has the one-parameter set of invariant measures parametrized by the…
It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…
Using our T1 theorem with an energy side condition allowing common point masses, we extend our previous work in arXiv:1310.4484v3 on one measure supported on a line, to include regular C(1,delta) curves and to permit common point masses. In…
Inference of the conditional dependence structure is challenging when many covariates are present. In numerous applications, only a low-dimensional projection of the covariates influences the conditional distribution. The smallest subspace…
We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…
It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact…
This correspondence studies an estimator of the conditional support of a distribution underlying a set of i.i.d. observations. The relation with mutual information is shown via an extension of Fano's theorem in combination with a…
The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_{\tau},\, 0<\tau\leq 1$, are studied. Unlike the case of the standard sphere ($\tau=1$), where there are no stable compact minimal…
In this note we study a minimization problem for a vector of measures subject to a prescribed interaction matrix in the presence of external potentials. The conductors are allowed to have zero distance from each other but the external…
We adopt a measure-theoretic perspective on the Riemannian approximation scheme proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds. We show that the zero-order term in the limit is a singular measure…
We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions…
We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on…
We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…