Local Geometry of Real and Complex Analytic Mappings

我的工作是研究解析几何和交换代数，这是现代数学的两个主要分支，在科学和工程中都有应用。它处理多项式函数和解析函数的分析以及它们所在的空间的几何。我的研究计划的主要主题是研究映射的局部退化(或奇点)。这种微小的映射退化可以完全破坏它们所应用到的对象的几何正则性。我的程序的主要目标之一是建立解析和多项式映射的奇点的可计算分类，重点是计算机代数的适用性。奇点理论的应用超出了数学范畴，例如在地震预报方面。我工作的另一个分支是一个理论，它将让我们更好地理解复分析中偏微分方程组的几何。这样的方程在科学和工程中被用来描述电磁和许多其他自然现象。这项研究为研究生和博士后研究员的培训提供了机会，并涉及到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.