Mathematical Sciences: Topological Properties of Singularities and Nonlinear Problems

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In singularity theory, topological methods have come to play an important role, both in the classification of singularities, as well as in determining the properties of mappings. In turn, these topological properties can be used to answer fundamental questions about nonlinear problems. Professor Damon has been developing topological analogues of the infinitesimal methods of Mather for answering questions about the topological classification of singularities. He has also discovered that the methods used for these results are closely related to quite different questions concerning the topology of discriminants and symmetry breaking problems in bifurcation theory. He hopes to develop further results for topological equivalence questions, to use recent developments to explore topology of discriminants and symmetry breaking for compact Lie groups, and to investigate the application of classification results for singularities of solutions of differential equations. Singularity theory has obvious relevance to other areas. Some of the most interesting solutions to systems of equations, both ordinary and differential, are the singular ones. In applications, it is often impossible to confine attention to the regular solutions to problems, and knowledge about the nature of singularities and about symmetry breaking is fraught with physical significance.