Related papers: On the geometry of sharp minima

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $\rho$-invariant. As an application our…

We investigate the geometry and topology of submanifolds under a sharp pinching condition involving extrinsic invariants like the mean curvature and the length of the second fundamental form. Several homology vanishing results are given.…

Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of…

This paper investigates the tilt stability of local minimizers for nonlinear programs under the relaxed constant rank constraint qualification in finite dimensions. By employing a neighborhood primal-dual approach and extending calculus…

We provide a simple proof of the radial symmetry of any nonnegative minimizer for a general class of quasi-linear minimization problems.

This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It…

This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular,…

We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the…

This note describes the construction of c U p-invariant differential operators on statistical manifolds, i.e. of operators canonically associated to a geometry which synthetizes the properties of conformal and projective geometries.

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like…

The purpose of the paper is to give a complete characterization of the continuity of lower envelopes in the infinite dimensional spaces in terms of the notion of c-regularity. As an application we introduce a variational unconstrained…

In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets. As an…

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities…

We show that there are minimal graphs in R^{n+1} whose intersection with the portion of the horizontal hyperplane contained in the unit ball has any prescribed geometry, up to a small deformation. The proof hinges on the construction of…

A surface M is called p-minimal if one of the coordinate functions is p-harmonic in the inner metric. We show that in the twodimensional case the Gaussian map of such surfaces is quasiconformal. In the case when the surface is a tube we…

Spin glass theory studies the structure of sublevel sets and minima (or near-minima) of certain classes of random functions in high dimension. Near-minima of random functions also play an important role in high-dimensional statistics and…

The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these…

Despite extensive study, the significance of sharpness -- the trace of the loss Hessian at local minima -- remains unclear. We investigate an alternative perspective: how sharpness relates to the geometric structure of neural…

In this paper we consider properties of medians as they pertain to the continuity and vanishing oscillation of a function. Our approach is based on the observation that medians are related to local sharp maximal functions restricted to a…