Exotic Topology and Geometry

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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).