Mathematical Sciences: Topological Classification of Singularities and Nonlinear Problems

John Mather用Rene Thom提出的思想回答了一些关于光滑映射奇点的基本问题。这种Thom-Mather方法通过将问题归结为可用代数求解的无穷小问题来解决问题。遗憾的是，这种方法不能应用于类似的拓扑问题。詹姆斯·达蒙一直在开发一种与Thom-Mather方法类似的拓扑结构。这使他能够在许多情况下获得拓扑学问题的答案。同样，他一直在应用奇点理论的结果来获得关于非线性微分方程解的结果。他打算进一步发展奇点理论中的拓扑等价的思想，研究符号计算在解决这些问题中的应用，并进一步探索奇点理论在非线性微分方程中的应用。

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.