Local geometry of real and complex analytic mappings

我的工作是在解析几何和交换代数的接口，在科学和工程应用的现代数学的两个主要分支。它们处理多项式和解析函数的分析以及它们所处空间的几何。我目前的研究课题是研究映射中的局部简并。这种微小的简并可以完全破坏由映射定义的对象的几何规则，因此知道如何检测它们是很重要的。我的程序的主要任务是建立简单的代数准则来控制映射的局部行为。这项研究为培养研究生和博士后提供了机会，并涉及与其他数学家的国际合作。

My work is at the interface of analytic geometry and commutative algebra, two major branches of modern mathematics with applications in science and engineering. They deal with the analysis of polynomial and analytic functions and the geometry of the spaces on which they live. The subject of my current research program is the study of the local degeneracies in mappings. Such minute degeneracies can completely destroy the geometric regularity of the objects defined by the mappings, and hence it is important to know how to detect them. The main task of my program is to establish simple algebraic criteria to control the local behaviour of a mapping. This research provides opportunities for the training of graduate students and postdoctoral fellows and involves international collaboration with other mathematicians.