Research in Geometry and Topology

这个项目的目标是理解三维庞加莱对偶群，克莱因群，平坦共形结构，对称空间中的多边形的结构，并将后者应用于表示论。 这个项目的部分重点是在几何，代数和物理中自然产生的物体的对称性结构。这种对称性既包括离散对称性（如墙纸图案和规则固体的对称性），也包括连续对称性（如空间的平移和旋转）。“负曲率”的存在（例如，这意味着三角形中的角之和可以小于180度）使得对这种对称性的研究更加有趣和具有挑战性。多边形提供了几何和李理论之间的联系，李理论是一个涉及连续和离散对称的数学领域。 多边形，这是熟悉的对象从高中几何，变得更加复杂时，考虑在负弯曲的空间。在这样的空间中多边形的“边长”变成矢量而不是数字。这样的多边形在李理论中提供了一个贫乏的工具。

The goal of this project is to understand structure of 3-dimensionalPoincare duality groups, Kleinian groups, flat conformal structures, polygons in symmetric spaces and apply the latter to the representation theory. This project focuses in part on the structure of symmetries of objectsnaturally arising in geometry, algebra and physics. Such symmetries include both discrete symmetries (like the ones of wallpaper patterns and regular solids) as well as continuous symmetries (like translations and rotations of the space). Presence of ``negative curvature'' (which implies for instance that sum of angles in a triangle can be less than 180 degrees) makes study of such symmetries more interesting and challenging. Polygons provide a link between geometry and Lie theory, which is a field of mathematics which involves both continuous and discrete symmetries. Polygons, which are familiar objects from the high-school geometry, become much more complex when considered in negatively curved spaces. The ``side-lengths'' of polygons in such spaces become vectors rather than numbers. Such polygons provide a poverful tool in the Lie theory.