Exotic Topology and Geometry

主要研究员（PI）在过去几年中的研究主要集中在奇异拓扑及其在几何中的应用。Farrell，在拓扑学中使用外来元素（例如，外来可微或PL结构，伪同位素空间的同伦群的非零性......等）产生结果微分几何（例如调和映射，Ricci流，空间的）。先验，这些结果似乎很难获得不使用存在的异国情调topological entities.This research sheds some light on the international fashion in whichTopology，Geometry and Analysis are interwoven.Paragraph #2：Topology studies properties of objects that are preserved underdeformations.在过去的50年里，人们发现了许多奇异的拓扑实体。我们用这些奇异的实体来说明几何学中某些众所周知的解析变形是有局限性的。这些分析变形类似于使用热方程获得的变形，热方程规定了物体上从零时刻（初始分布）到无穷大（最终平衡）的热量分布。

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).