Local Geometry of Real and Complex Analytic Mappings

实数和复数解析映射的局部几何

• 批准号: 355418-2013
• 负责人: Adamus, Janusz
• 金额: $ 1.38万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572571
• 关键词: Local; Geometry; Real; Complex; Analytic

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.

我的工作是在解析几何和交换代数的接口，现代数学的两个主要分支与应用在科学和工程。它涉及分析多项式和解析函数和几何空间上，他们的生活。我的研究计划的主要课题是研究局部退化（或奇异）的映射。映射的这种微小退化可以完全破坏它们所应用的对象的几何规则性。我的计划的主要目标之一是建立一个可计算的分类奇异的解析和多项式映射，强调计算机代数的适用性。奇点理论的应用超出了数学，例如在预测地震。我的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的偏微分方程几何。这些方程在科学和工程中被用来描述电磁学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，并涉及与其他数学家的国际合作。