A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature at least k is an Alexandrov's space of curvature at least k. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo.
Gradient-based optimization methods are the most popular choice for finding local optima for classical minimization and saddle point problems. Here, we highlight a systemic issue of gradient dynamics that arise for saddle point problems, namely the presence of undesired stable stationary points that are no local optima…
Gradient perturbation, widely used for differentially private optimization, injects noise at every iterative update to guarantee differential privacy. Previous work first determines the noise level that can satisfy the privacy requirement and then analyzes the utility of noisy gradient updates as in the non-private cas…
We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed \emph{coarse scalar curvature}, generalizes a recent definition of Ricci curvature for…
We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates.…
In this paper, we prove that if the initial submanifold M0 of dimension n(≥6) satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for subma…
The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativ…
The study optimizes cell membranes' shapes based on curvature and proves existence of minimizers.
problem Optimizing cell membranes' shapes with respect to curvature.
method Modeling cell membranes as optimal shapes with L2-deficit of mean curvature to spontaneous curvature, and proving lower semi-continuity and existence of minimizers.
result Smoothly embedded minimizers and diameter bounds are obtained.
In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed thre…
This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure …
Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature.
The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth n-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called p-Ricci curvature which corresponds to taking the trace of the Ri…
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
The study finds a diameter bound for graphs with positive entropic Ricci curvature, with optimal bounds for arithmetic mean.
problem Finding diameter bounds for graphs with positive entropic Ricci curvature.
method Using a localized gradient estimate and an equivalent definition of entropic Ricci curvature, the study derives a Bonnet-Myers type diameter bound.
result The derived diameter bound is optimal for arithmetic mean, but not for logarithmic mean.
We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval [0,T) can be extended over time T. Moreover, we show that the condition is optimal in some sense.
Using δ-invariants and Newton--Okounkov bodies, we derive the optimal volume upper bound for Kähler manifolds with positive Ricci curvature, from which we get a new characterization of the complex projective space.