Geometry and Dynamics in Riemannian and Finsler Spaces

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

Abstract for DMS - 0103739 The project can be conditionally divided into the following (related) parts: Study of periodic metrics (including area-minimizing properties of flats in normed spaces, symplectic filling volumes for Finsler metrics, asymptotic volume growth of Finsler tori, Riemannian metrics without conjugate points on products, and Lagrangian systems on tori without conjugate points); Relationship between bi-Lipschitz equivalence and quasi-isometries (with the most intriguing cases of general Penrose tilings and finitely presented groups, including co-compact lattices in the same Lie group); Products of non-commuting maps, flows of positive metric entropy, and sequential dynamics; Applications of geometry of non-positive curvature to algorithmics and dynamics; Geometry of non-negatively-curved manifolds (foliations by minimal surfaces, isolated flat totally geodesic tori); the PI's graduate students work on generalizations of the Finite Distance Theorem to non-Abelian groups, approximations of embedded surfaces with small variation of Gaussian curvature by developing surfaces, constructing Lipschitz homeomorphisms with prescribed Jacobians, generating certain groups by products of conjugates of elements from a bounded subset. The first part of the project deals with large-scale invariants of periodic metrics. Their physical analogs are macroscopic properties of periodic media (such as a crystal substance), and the problem is to understand how such properties can be recovered from microscopic characteristics and vice versa. A large part of the project belongs to a borderline between geometry and dynamics, and in particular new applications of geometric methods. For instance, problems of stability in sequential dynamics model situations where the laws of evolution of an object (for instance, a physical or an ecological system) are subject to small perturbations; it is desirable to understand the result of such perturbations in the large time scale. There are also applications of modern geometry of singular spaces to problems originated from statistical physics (such as estimates on the number of collisions of particles in gas models, a problem that goes back to Boltzmann), and to computational problems (such as: how to numerically find a shortest path between around several obstacles). The last part of the project deals with stability of geometric objects described by curvature-type characteristics. Indeed, whenever we study a geometric object (for instance, a surface), we deal with imprecise information. Thus it is important to understand whether small deviations in this information can result in crucial changes for the geometric object (or even a non-existence of a model object).