Geometry and Dynamics in Riemannian and Finsler Spaces

Abstract

Award: DMS-0412166

Principal Investigator: Dmitri Burago

D. Burago proposes to continue his work on a number of long-term

projects in Riemannian and Finsler geometry, dynamical systems of

geometric origin, and geometric group theory. The projects

include: geometry of periodic metrics, area-minimizing surfaces

in normed spaces and minimal fillings, and ellipticity of surface

area functionals; low dimensional partially hyperbolic

diffeomorphisms; unbounded bi-invariant quasi-semi-norms and

'large' groups; manifolds without conjugate points; applications

of singular geometry to dynamics of billiard systems and certain

algorithmic problems of geometric origin; and approximations by

PL-isometries. Among the problems the proposal is aimed at there

are: Busemann's Conjecture that flats are area-minimizing

surfaces in normed spaces; styding the structure of the class of

partially-hyperbolic systems; finding weak versions of Hofer's

norm (possibly on some groups of volume-preserving

homeomorphisms); various generalizations of E.Hopf's conjecture

on tori without conjugate points. This project continues the

proposer's previous research, including a solution of the E. Hopf

conjecture on tori without conjugate points posed by Hedlund and

Morse in the 40s, a 'Boltzman-Sinaj' problem on the existence of

uniform estimates on the number of collisions in hard ball gas

models, the two-dimensional case of H. Busemann's problem

mentioned above, H. Furstenberg's problem on the existence of

bi-Lipschits non-equivalent separated nets and J. Moser's problem

on the existence of a continuous function that is not a Jacobian

determinant of a Lipschits homeomorphism (all from the 60s). The

conjectures and directions of research suggested in the proposal

grew from ideas and methods developed by the proposer and his

collaborators while working on these problems.

Even though most topics of the proposal belong to rather abstract

areas of mathematics, their motivations lie in real-word

problems. The physical analogs of large-scale invariants of

periodic metrics are macroscopic properties of periodic media

(such as crystal substances: i.e., the rate of propagation of

radiation etc), and one wants to relate these properties to

microscopic characteristics. Hyperbolic dynamics is really well

understood, and it forms the first and the simplest example of

chaotic models; however, little is know about partially

hyperbolic systems, which offer a much more realistic model; the

project is aimed in giving new insight into such systems with a

small number of degrees of freedom. The 'Boltzman-Sinaj' problem

on the existence of uniform estimates on the number of collisions

in hard ball gas models originated from the most basic research

in statistical physics. Study of the geodesic flows, billiard

systems, geometric complexity, and optimal strategies may result

in better understanding of (stability) of certain models in

thermodynamics, biology, sociology, and physics, especially when

dealing with imprecise data, and perhaps result in new

computational algorithms.