Mathematical Sciences: Geometric Structures and Discrete Groups

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9504764 Goldman The proposal deals with various classical geometries and their discrete automorphism groups: complex hyperbolic geometry, spherical Cauchy-Riemann geometry, flat Lorentzian geometry, conformal geometry and real projective geometry are to be studied. These geometries are all generalizations of the geometry of the hyperbolic plane, yet they lack totally geodesic hypersurfaces making it difficult to build fundamental polyhedra - one can construct discrete automorphism groups directly from fundamental polyhedra. A related topic dealing with representation spaces of surface groups is also contained in the proposal: these spaces can be identified with moduli spaces of holomorphic vector bundles over Riemann surfaces, hence have ramifications in algebraic geometry and mathematical physics. Hyperbolic geometry is the oldest known non-Euclidean geometry, and possesses many interesting properties not encountered in Euclidean geometry. Discrete automorphism groups are closely related to regular polyhedra, and finding them in a non-Euclidean geometry can often be difficult. Collections of these groups are also related to certain gauge theories arising in mathematical physics.