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。它处理多项式和解析函数的分析以及它们所在空间的几何。我的主要研究课题是映射的局部简并（或奇点）的研究。这种映射的微小退化可以完全破坏它们所应用对象的几何规则。我的程序的主要目标之一是建立一个可计算的分析和多项式映射的奇点分类，重点是计算机代数的适用性。奇点理论在数学之外也有应用，例如在地震预报中。我工作的另一个分支是一个理论，它将使我们更好地理解复分析中偏微分方程的几何。这些方程在科学和工程中用来描述电磁和许多其他自然现象。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. It deals with the analysis of polynomial and analytic functions and the geometry of spaces on which they live. The main subject of my research program is the study of local degeneracies (or singularities) of mappings. Such minute degeneracies of mappings can completely destroy the geometric regularity of objects to which they are applied. One of the main goals of my program is to establish a computable classification of singularities of analytic and polynomial mappings, with the emphasis on applicability of computer algebra. Theory of singularities has applications beyond mathematics, for example in forecasting of earthquakes. Another ramification of my work is a theory that will give us a better understanding of the geometry of partial differential equations in complex analysis. Such equations are used in science and engineering to describe electromagnetism and many other natural phenomena. This research provides opportunities for the training of graduate students and postdoctoral fellows and involves international collaboration with other mathematicians.