Mathematical Sciences: Differential Topology and Dynamics ofGroup Actions and Foliations

[401688] Hurder开完全流形几何性质的研究是纯数学中一个迅速发展的研究领域。Hurder的研究项目旨在通过探索“无穷远处的结构”和空间的渐近对称性，对开放流形上的几何和动态现象有更深的理解。他的研究结合了粗糙上同调、循环上同调、建筑、动力学和狄拉克型算子的谱理论等技术。近年来，作用渐近行为的结构理论变得越来越复杂，是解决一些众所周知的关于开流形和这些空间上对称群的分类问题的一种新兴技术选择。在这项工作中解决的问题也与数学物理和解释自然现象密切相关，特别是将开放空间上的渐近结构与几何算符的谱理论联系起来的工作。该项目的其他方面目前正在应用于分析平铺图的计算机模拟和二维和三维空间中的随机运动动力学，最终应用于解释气体的行为（高维台球问题）。＊＊＊

9401688 Hurder The study of geometric properties of open complete manifolds is a rapidly developing area of research in pure mathematics. Hurder's research program seeks to gain a deeper understanding of geometric and dynamical phenomenon on open manifolds by exploring the "structures at infinity" and asymptotic symmetries of spaces. His investigation combines techniques from coarse cohomology, cyclic cohomology, buildings, dynamics and spectral theory of Dirac-type operators. The structure theory for the asymptotic behavior of actions has become much more sophisticated in recent years, and is an emerging technology of choice for addressing some well-known problems regarding the classification of open manifolds and the symmetry groups on these spaces. The problems addressed in this work are also closely tied to mathematical physics and explaining natural phenomenon, especially the work relating asymptotic structures on open spaces to the spectral theory of geometric operators. Other aspects of the project are currently being applied to the analysis of computer simulations of tiling maps and random dynamics of motions in two and three space, with eventual applications to explaining the behavior of gases (the higher dimensional billiard problem). ***