Mathematical Sciences: Spectral Asymptotics for Hyperbolic Surfaces and Real-Projective Structures

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9504176 Wolpert The investigator proposes to continue his investigations of the large eigenvalue asymptotics of hyperbolic Riemann surfaces. He proposes to consider the question of unique quantum ergodicity for the classical modular group, connections between spectral asymptotics and questions in number theory, and the geometry of the moduli space of real projective structures for a compact surface. Riemann surfaces can be thought of as multi-sheeted covers of the plane; have interesting topological properties. In addition, Riemann surfaces play a crucial role in the study of complex-valued functions (the graph of a complex-valued function is a surface in a 4-dimensional space). Indeed they were originally constructed as natural domains for complex-valued functions.