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

该奖项支持加州大学圣克鲁斯分校的Svetlana Katok教授对局部对称空间和负曲率流形的几何和分析的研究。Katok博士的项目包括两个主要部分：首先，她打算研究紧致负曲面上测地线流动的上同调方程；其次，她将把她已经证明的关于Fuchsian群上自同构形式的生成定理推广到大于2维的局部对称空间。非欧几里得平面几何开始于19世纪初，当时只是数学上的一种好奇，但到19世纪末，数学家们已经意识到，许多具有根本重要性的物体在其基本性质上都是非欧几里得的。对非欧几里得平面几何的详细研究产生了现代数学的几个分支，其中模和自同构形式的研究是最活跃的分支之一。这一领域主要涉及关于整数的问题，但在使用几何和分析方面，如Katok教授的作品所示，它保留了与其历史根源的联系。

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.