Exotic Topology and Geometry

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