Geometry and Dynamics in Riemannian and Finsler Spaces

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AbstractAward: DMS-0412166Principal Investigator: Dmitri BuragoD. Burago proposes to continue his work on a number of long-termprojects in Riemannian and Finsler geometry, dynamical systems ofgeometric origin, and geometric group theory. The projectsinclude: geometry of periodic metrics, area-minimizing surfacesin normed spaces and minimal fillings, and ellipticity of surfacearea functionals; low dimensional partially hyperbolicdiffeomorphisms; unbounded bi-invariant quasi-semi-norms and"large" groups; manifolds without conjugate points; applicationsof singular geometry to dynamics of billiard systems and certainalgorithmic problems of geometric origin; and approximations byPL-isometries. Among the problems the proposal is aimed at thereare: Busemann's Conjecture that flats are area-minimizingsurfaces in normed spaces; styding the structure of the class ofpartially-hyperbolic systems; finding weak versions of Hofer'snorm (possibly on some groups of volume-preservinghomeomorphisms); various generalizations of E.Hopf's conjectureon tori without conjugate points. This project continues theproposer's previous research, including a solution of the E. Hopfconjecture on tori without conjugate points posed by Hedlund andMorse in the 40s, a "Boltzman-Sinaj" problem on the existence ofuniform estimates on the number of collisions in hard ball gasmodels, the two-dimensional case of H. Busemann's problemmentioned above, H. Furstenberg's problem on the existence ofbi-Lipschits non-equivalent separated nets and J. Moser's problemon the existence of a continuous function that is not a Jacobiandeterminant of a Lipschits homeomorphism (all from the 60s). Theconjectures and directions of research suggested in the proposalgrew from ideas and methods developed by the proposer and hiscollaborators while working on these problems.Even though most topics of the proposal belong to rather abstractareas of mathematics, their motivations lie in real-wordproblems. The physical analogs of large-scale invariants ofperiodic metrics are macroscopic properties of periodic media(such as crystal substances: i.e., the rate of propagation ofradiation etc), and one wants to relate these properties tomicroscopic characteristics. Hyperbolic dynamics is really wellunderstood, and it forms the first and the simplest example ofchaotic models; however, little is know about partiallyhyperbolic systems, which offer a much more realistic model; theproject is aimed in giving new insight into such systems with asmall number of degrees of freedom. The "Boltzman-Sinaj" problemon the existence of uniform estimates on the number of collisionsin hard ball gas models originated from the most basic researchin statistical physics. Study of the geodesic flows, billiardsystems, geometric complexity, and optimal strategies may resultin better understanding of (stability) of certain models inthermodynamics, biology, sociology, and physics, especially whendealing with imprecise data, and perhaps result in newcomputational algorithms.