Mathematical Sciences: Topological Properties of Singularities and Nonlinear Problems

在奇点理论中，拓扑方法在奇点的分类以及确定映射的性质方面都扮演着重要的角色。反过来，这些拓扑性质可以用来回答有关非线性问题的基本问题。达蒙教授一直在开发Mather的无穷小方法的拓扑类比，用于回答有关奇点的拓扑分类的问题。他还发现，用于这些结果的方法与分歧理论中关于判别式拓扑和对称破缺问题的完全不同的问题密切相关。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紧李群的判别式的拓扑和对称破缺，并研究微分方程解的奇性分类结果的应用。奇点理论与其他领域有明显的相关性。常微分方程组的一些最有趣的解是奇异解。在应用中，通常不可能将注意力局限于问题的正则解，而关于奇点的性质和对称破缺的知识充满了物理意义。

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.