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We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality $C_D \leq C_N$ between…

Mirror Descent (MD) is a scalable first-order method widely used in large-scale optimization, with applications in image processing, policy optimization, and neural network training. This paper generalizes MD to optimization on Riemannian…

Here we introduce reflection positive doubles, a general framework for reflection positivity, covering a wide variety of systems in statistical physics and quantum field theory. These systems may be bosonic, fermionic, or parafermionic in…

Self-testing is a phenomenon where the use of specific quantum states or measurements can be inferred solely from the correlations they generate. We introduce a universal method for conducting robustness analysis in the self-testing of…

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat…

We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The…

We develop a general deformation principle for families of Riemannian metrics on smooth manifolds with possibly non-compact boundary, preserving lower scalar curvature bounds. The principle is used in order to strengthen boundary…

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented…

This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such…

We derive nonlinear stability results for numerical integrators on Riemannian manifolds, by imposing conditions on the ODE vector field and the step size that makes the numerical solution non-expansive whenever the exact solution is…

We prove an integral inequality and two stability results for the ADM mass on AE K\"ahler manifolds of all complex dimensions. The inequality bounds the ADM mass from below by an integral of the scalar curvature and the Hessian of certain…

Optimization on Riemannian manifolds widely arises in eigenvalue computation, density functional theory, Bose-Einstein condensates, low rank nearest correlation, image registration, and signal processing, etc. We propose an adaptive…

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is…

The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.

This work presents a thorough numerical study of Riemannian Newton's Method (RNM) for optimization problems, with a focus on the Grassmannian and on the Stiefel manifold. We compare the Riemannian formulation of Newton's Method with its…

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental forms: For a manifold $M^d$ endowed with a reference metric and a reference shape…

The tension field of the identity map from a statistical manifold to a Riemannian statistical manifold, which shares the same Riemannian metric, is the Tchevychev vector field multiplied by negative one. We derive a new class of statistical…

Estimating the rate of convergence of the empirical measure of an i.i.d. sample to the reference measure is a classical problem in probability theory. Extending recent results of Ambrosio, Stra and Trevisan on 2-dimensional manifolds, in…

We present a generalization of continuous position measurements that accounts for a spatially inhomogeneous measurement strength. This describes many real measurement scenarios, in which the rate at which information is extracted about…

We study optimization over Riemannian embedded submanifolds, where the objective function is relatively smooth in the ambient Euclidean space. Such problems have broad applications but are still largely unexplored. We introduce two…