Exotic Topology and Geometry

最近几年，主要研究者(PI)的研究主要集中在奇异拓扑学及其在几何学中的应用。PI与F.T.Farrell合作，利用拓扑学中的奇异元素(例如奇异可微结构或PL结构，伪同位素空间的同伦群的非零性等)来产生微分几何(如调和映射，Ricci流，度量空间)的结果。先验地，如果不使用奇异拓扑体的存在，这些结果似乎很难获得。这项研究揭示了拓扑学、几何学和分析相互交织的错综复杂的方式。第2段：拓扑学研究在变形下保持的对象的性质。在过去的50年里，人们发现了许多奇怪的、意想不到的“奇异”拓扑学。我们使用这些奇异的实体来证明某些众所周知的几何学中的解析变形是有局限性的。这些解析变形类似于使用热方程得到的变形，该方程规定了从时间零(初始分布)到无穷大(最终平衡)的物体上的热分布。

The research of the Principal Investigator (PI) in the last few years hasbeen centered mainly inExotic Topology and its applications to Geometry.The (PI), in collaboration with F.T. Farrell, use exotic elements inTopology (e.g. exotic differentiable or PL structures, non-vanishing ofthehomotopy groups of the space of pseudo-isotopies...etc) to produce resultsin Differential Geometry (e.g Harmonic Maps,Ricci Flow, Space of Metrics). A priori, these results seem difficult toobtain without the use of the existence of exotic topological entities.This research sheds some light on the intricate fashion in whichTopology, Geometry and Analysis are interwoven.Paragraph #2:Topology studies properties of objects that are preserved underdeformations. Many strange and unexpected "exotic" topological entitieswere discovered in the last 50 years. We use these exotic entities to showthat certain well know analytic deformations in geometryhave limitations. These analytic deformations are analogous tothe one obtained using the Heat Equation, which rules thedistribution of heat on a body, from time zero (initial distribution) toinfinity (final equilibrium).