Mathematical Sciences: Topological Classification of Singularities and Nonlinear Problems

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A number of fundamental questions about singularities of smooth mappings have been answered by John Mather using ideas proposed by Rene Thom. This Thom-Mather approach solves the problems by reducing them to infinitesimal problems which can be solved algebraically. Unfortunately, this approach cannot be applied to the analogous topological problems. James Damon has been developing a topological analogue of the Thom-Mather approach. This has enabled him to obtain answers to the topological questions in a number of circumstances. Likewise, he has been applying results from singularity theory to obtain results about solutions to nonlinear differential equations. He intends further to develop ideas on topological equivalence in singularity theory, to investigate the use of symbolic computations in addressing these questions, and to explore further the applications of singularity theory to nonlinear differential equations.