Applications of ergodic theory to geometry: Dynamical Zeta Functions and their applications

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The proposed research lies at the interface of Ergodic Theory and Dynamical Systems, geometry, number theory, partial differential operators and mathematical physics. Central to this research programme are the the application of ideas from smooth ergodic theory to problems in different areas of mathematics. As such it is a highly intra-disciplinary research program. It also seems very timely, since there has been an explosion of activity in these areas in the last year which has attracted widespread attention. The proposed research is at the cutting edge of this development. In particular, the basis for this project rests on four important inter-related strands in applications of ergodic theory and dynamical systems to other areas: zeta functions and Poincare series (with their connections to number theory and geometry); Decay of correlations and resonances (with applications to the physical sciences); Numerical algorithms (with applications to both Pure and Applied Mathematics); and Teichmuller theory and Weil-Petersson metrics (at the boundary of ergodic theory, analysis and geometry).The study of geometric zeta functions for closed geodesics on negatively curved manifolds was initiated by Fields Medallist A. Selberg in the 1950s (following his earlier work on number theory). Selberg studied the case of constant curvature manifolds, using trace formulae and ideas from representation theory which do not generalise. However, recent work of Giulietti, Liverani and myself used a completely different viewpoint involving ideas in ergodic theory to extend the zeta function for negatively curved manifolds (and even more generally smooth Anosov flows, generalizing the geodesic flow). This provides the starting point for our proposed research on zeta functions, providing both a springboard to a whole host of significant applications and providing the scientific framework via the new ideas and techniques it initiated.