Geometry and Dynamics in Riemannian and Finsler Spaces

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

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.