Geometry and Dynamics in Riemannian and Finsler Spaces

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The main projects of the proposed researchinclude: boundary rigidity and related inverse problems, minimal fillings, asymptotic geometry of tori and area-minimizing surfaces in normed spaces; study of partially hyperbolic diffeomorphisms; conjugation-invariant quasi-norms; ``area structures and spaces". They are aiming at solving a number of long-standing and important problems, and formulating new problems and new directions of research. The four projects that form the core of the proposal have already yielded a number of important results.Among the problems the proposal is aimed at are Michel's Conjecture that simple metrics are boundary rigid, Pu's conjecture on the filling area of the circle, Busemann's Conjecture that flats in normed spaces are area-minimizers, classifications of partially-hyperbolic systems, studying ``area structures", finding geometric (conjugation-invariant) and ``non-dynamical" (not asymptotic) invariants of diffeomorphisms preserving certain structures, and various generalizations of the E. Hopf Conjecture.Most of the conjectures and directions of research suggested in the proposal grew from ideas and methods developed by the PI in his previous research, and projects described in the proposal continue the research resulted in solving them.Many topics of the proposed research have rather feasible relation to applied science. The Boundary Rigidity Problem and related Inverse Problems are motivated by important problems in geophysics and medical imaging.To visualize that, imagine that one wants to find out what the Earth is made of.More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at somepoints of the surface of the body and "listen when the sound gets to other points".The question is if this information is enough to determine what is inside. The proposed research already resulted in the first result of this kind for a reasonably general case (with the restriction that the properties of the material do not change to much from point to point), and there is a cautions hope to handle the general case.The physical analogs of large-scale invariants of periodic metrics are macroscopic properties of periodic media (such as a crystal substance), and one wants to relate these properties to microscopic characteristics; similarly, study of partially hyperbolic systems, geodesic flows, and geometric complexity may result in better understanding of (stability) of certain models in thermodynamics, biology, sociology, and physics, especially when dealing with imprecise data.