Dynamics of Semi-Dispersing Billiards; Geometry of Operator Symmetric Spaces; Rigidity of Higher-Rank Anosov Actions

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AbstractFerlegerThe proposer will continue his research in the following three directions: applications of non-regular Riemannian geometry to the dynamics of billiard systems, rigidity of Anosov actions of higher rank abelian groups and Banach geometry of non-commutative symmetric spaces. Specifically, in his study of dynamics of semi-dispersing billiards on Riemannian manifolds, the proposer plan to address the following questions: Is it possible to give any estimate for the topological entropy of the first return map of a non-degenerate semi-dispersing billiard with a ``finite horizon''? What can be said about the topological entropy of the billiards on manifolds without the non-positive curvature restriction? Most fundamental is the question about existence of universal unfolding space for arbitrary non-degenerate semi-dispersing billiards on non-positively curved manifolds of dimension greater than three, that the proposer plans to attack using the machinery he (jointly with D.Burago and B.Kleiner) has developed recently. Continuing his (joint with A.Katok) study of cohomological rigidity of Anosov actions of higher-rank Abelian groups, the proposer will attempt to establish cohomological rigidity of standard Anosov actions different from Weyl chamber flow; to obtain infinitesimal tensor rigidity of lattice actions on Furstenberg boundaries and differentiable rigidity of the projective actions of cocompact lattices. Central for proposer's program of studying the geometry of noncommutative operator spaces is certain generalization of Hilbert transformation and upper triangular projection at the same time, a spectral projection associated with a strongly continuous representation of a compact Abelian group on an arbitrary (UMD)-space. The prevalent theme and tool in the proposal is the theory of dynamical systems. The theory in general investigates the evolution of a physical or mathematical system over time. Theory of dynamical systems is intertwined closely with most of the major areas of mathematics and it often takes its basic examples from and applies its results to such different fields as physics, engineering, biology or sociology. Theory of billiard systems, for instance, lies at the interface of geometry on one hand and dynamics and theoretical mechanics on the other. Classical thermodynamics is essentially nothing else but mechanics of huge billiard systems. This is why a number of problems of the theory were first posed by physicists in their attempt to "justify" thermodynamics. Such justifications still is just a dream; but the proposer believes that his research brings us a little closer to its realization. One of these problems that has been open for almost half of the century (about existence of an estimate on the number of collisions between molecule of a gas) recently was solved with help of geometry of somewhat pathological spaces. Discovery of such an unexpected connection opened up several opportunities in the area that the proposer would like to pursue as outlined above.