Mathematical Sciences: Topological Classification of Singularities and Nonlinear Problems

关于奇点的一些基本问题 光滑映射已经被约翰·马瑟用思想回答了 由Rene Thom提出。 这种方法解决了 将问题简化为无穷小问题， 用代数方法解决。 不幸的是，这种方法不能 适用于类似的拓扑问题。 詹姆斯·达蒙一直在开发一种拓扑类似物， 马瑟的方法 这使他能够获得答案 在很多情况下都能解决拓扑问题。 同样，他一直在应用奇点理论的结果， 得到非线性微分方程解的结果 方程 他打算进一步发展思想拓扑 奇点理论中的等价性，以研究 符号计算在解决这些问题，并 进一步探索奇点理论在 非线性微分方程

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.