Geometry in Teichmüller and moduli spaces

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Riemann surfaces are ubiquitous throughout mathematics. They appear in algebra as solutions to polynomial equations, in complex analysis as natural domains for analytic continuation, and in geometry as quotients of the hyperbolic plane by discrete groups of isometries. While a single Riemann surface can be quite beautiful (e.g. Klein's quartic), the purpose of Teichmuller theory is to study them in families.******The moduli space of a closed surface S is the set of Riemann surfaces homeomorphic to S up to conformal equivalence. Since some surfaces have more symmetries than others, that space has a complicated structure: it is not a manifold but an orbifold (think manifold with corners). To simplify matters, one can keep track of additional topological data. A point in Teichmuller space is thus a Riemann surface together with a homotopy class of homeomorphism to S, called a marking. This space is simpler than moduli space: it is homeomorphic to Euclidean space of dimension 6g - 6, where g is the genus of S. The Teichmuller distance between two points in Teichmuller space mesures how far the two surfaces are from being conformally equivalent, in terms of how much angles need to be distorted to go from one surface to the other. The mapping class group of homotopy classes of orientation-preserving homeomorphisms of S acts on Teichmuller space by change of marking. Since this action preserves Teichmuller distance, the latter descends to a metric on the quotient, which is moduli space. ******In other words, the space of all Riemann surfaces of a given topological type has a shape and a geometry of its own. That geometry has been studied extensively by Ahlfors, Bers, Royden, Mumford, Thurston, Masur, Minsky, McMullen, Mirzakhani and many others, with applications to Kleinian groups, complex dynamics and topology. The goal of the present research program is to further understand that geometry, and to compute and visualize it. One specific objective is to find enough totally geodesic submanifolds in Teichmuller space to construct compact convex sets with non-empty interior.