Mathematical Sciences: Some Problems in Analysis on Locally Symmetric Spaces and Manifolds of Negative Curvature

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This award supports the research on the geometry and analysis of locally symmetric spaces and manifolds of negative curvature of Professor Svetlana Katok of the University of California at Santa Cruz. Dr. Katok's project consists of two main parts: first, she intends to study the cohomological equations associated with geodesic flows on compact negatively curved surfaces; and second, she will generalize spanning theorems that she has already proved for automorphic forms on Fuchsian groups to locally symmetric spaces of dimension greater than two. Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of modular and automorphic forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, as exemplified by the work of Professor Katok, it retains connection to its historical roots.