Mathematical Sciences: Differential Topology and Dynamics ofGroup Actions and Foliations

小行星9401688 开完备流形的几何性质的研究是纯数学中一个迅速发展的研究领域。 赫德的研究计划旨在通过探索空间的“无穷大结构”和渐近对称性，更深入地了解开流形上的几何和动力学现象。 他的研究结合了粗糙上同调，循环上同调，建筑物，动力学和Dirac型算子谱理论的技术。 近年来，作用的渐近行为的结构理论已经变得更加复杂，并且是解决关于开流形分类和这些空间上的对称群的一些众所周知的问题的新兴技术。 在这项工作中解决的问题也密切相关的数学物理和解释自然现象，特别是工作有关的渐近结构的开放空间的谱理论的几何算子。 该项目的其他方面目前正在应用于分析平铺地图的计算机模拟以及二维和三维空间中运动的随机动力学，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). ***