Geometry and Dynamics in Riemannian and Finsler Spaces

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).