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

该奖项支持对几何学的研究， 局部对称空间和负流形的分析 曲率教授斯维特拉娜Katok的大学 加州的圣克鲁斯。卡托克博士的项目包括两个 主要部分：首先，她打算研究的上同调 紧负空间上测地线流方程 曲面;其次，她将概括跨越 定理，她已经证明了自守形式上 Fuchsian群到高维局部对称空间 比两个多。 非欧平面几何始于19世纪初 世纪作为一个数学的好奇心，但在年底， 世纪，数学家们已经意识到， 基本的重要性是非欧几里德在其基本性质。 对非欧平面几何的详细研究， 上升到现代数学的几个分支，其中 模和自守形式的研究是最活跃的研究之一。 此字段主要涉及有关 整数，但在其使用的几何和分析，作为 以卡托克教授的研究为例，它保留了 它的历史根源。

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.