Metrics, Measures, and Identities on Moduli Spaces

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The PI studies the interconnections between the fields of geometry, physics, dynamics, statistics and number theory. A geometry on a space is a means of measuring distance in the space and can be thought of as giving the space a shape. One approach to studying the geometry on a space is to consider the geodesic flow; this is an object that generalizes the notion of a straight line to spaces that are curved. By studying the properties of this object much can be discovered about the geometry itself. For example, the question of how many flow paths close up is related to the distribution of prime numbers. Often different geometries can be placed on a space and thus one obtains a space of shapes. A natural question to ask is if this space of shapes can be given a nice shape itself. The PI proposes to study these geometries on the space of geometries and investigate their properties. The PI will continue his commitment to both undergraduate and graduate education. The PI will mentor graduate students and postdoctoral assistant professors on research related to the project. The PI will also give research talks, expository talks, minicourses and lecture series on material related to the proposal as well as organize conferences. The research plan of the PI centers around the use of certain geometric measures to define structures on moduli spaces and representation varieties. Such measures include the Hausdorff measure on the limit set of a Kleinian group, geodesic currents, the Patterson-Sullivan measure of a Kleinian group, equilibrium measures defined using Themodynamics associated with representations of hyperbolic groups, and push-forwards of volume measures by certain geometrically defined functions. One area of study is Higher Teichmuller Theory which is the study of representation spaces of hyperbolic groups into semi-simple Lie groups. These are generalizations of the classical Teichmuller space. Using Thermodynamics, the PI and collaborators define a Pressure geometry on this Higher Teichmuller space. space. The PI proposes to study the geometric property of this metric including its curvature, metric completion, and isometry group. Another area of proposed study is geometric identities; these are equations that hold on a moduli space of geometries. The PI and collaborators derive such identities by studying the statistical properties of the geodesic flow on a hyperbolic manifold. The PI proposes to study these identities and their relation to other known identities.