Mathematical Sciences: Topological Properties of Singularities and Nonlinear Problems

在奇点理论中，拓扑方法开始发挥作用， 一个重要的角色，无论是在奇点的分类， 以及确定映射的属性。 反过来， 这些拓扑性质可以用来回答 关于非线性问题。 达蒙教授 发展无穷小方法的拓扑类似物， Mather回答有关拓扑的问题 奇点的分类 他还发现， 用于这些结果的方法与相当密切相关 关于判别式拓扑的不同问题， 分叉理论中的对称破缺问题。 他希望 进一步发展拓扑等价问题的结果， 利用最近的发展来探索判别式的拓扑结构， 紧李群的对称破缺，并研究 奇异性分类结果的应用 微分方程的解 奇点理论与其他领域有着明显的相关性。 一些最有趣的方程组解， 普通的和微分的，都是奇异的。 在 应用程序，往往不可能将注意力集中在 问题的常规解决方案，以及关于 奇点和对称性破缺充满了 物理意义。

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.