CAREER: Moduli of curves via topology, geometry, and arithmetic

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In the study of geometry, one often gains insight into the nature of a particular geometric object by studying all such objects simultaneously. For instance, one might find the optimal configuration of robots in a factory by studying the "space" of all such configurations of robots, as the geometry of the latter can lend insight as to which particular configurations are problematic in a given situation (such as when robotic arms might jam or interfere with each other). Such "spaces" of geometric structures are called moduli spaces. The investigator's project is dedicated to a study of moduli spaces of surfaces, a key feature of the work being the interaction of geometry and algebra in the study of moduli. Some of this work will be conducted with the assistance of graduate students. The investigator will develop a graduate topics course in moduli spaces to stimulate interaction between students working in geometry and topology and those working in algebra. The investigator will also run biennial workshops designed to foster the upward professional development of graduate and undergraduate students working in fields related to moduli as well as to stimulate interaction across barriers between these disciplines.The investigator will study the moduli of Riemann surfaces. The proposed research naturally falls into three projects. The first is concerned with profinite aspects of mapping class groups of surfaces. In particular, the investigator will further develop profinite Teichmueller theory as initiated by Boggi, and will also pursue a study of centralizers in profinite completions of mapping class groups. The second project concerns the geometry of surface bundles and their relation to topology and algebra. In particular, the investigator will study distinctions between a number of geometric properties of subgroups of mapping class groups in analogy with more classical work in Kleinian groups, and will also continue a search for atoroidal surfaces bundles over surfaces. The third portion of the proposed work is dedicated to a study of the deformation theory of hyperbolic 3-manifolds. In particular, the investigator will continue study of certain analytic functions between Teichmuller spaces introduced by Thurston in his approach to his Geometrization conjecture, which come to bear on problems related to making Geometrization effective, as well as toward understanding certain pathology in the study of general deformation spaces of 3-manifolds.