Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.

奇点的解决及其应用。

• 批准号: 8949-2013
• 负责人: Milman, Pierre
• 金额: $ 2.77万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=521202
• 关键词: Resolution; Singularities; its; applications.; Analysis

Singularities express irregularities of form in many branches of mathematics in a way similar to what it means in an everyday language and are a basic object of study in most of the mathematics and its applications. The important features of form are often concentrated at singularities. The objective of my research is to find links between the information encoded in the geometry of and the analysis on singular objects.This led me to a discovery of fundamental links between algebraic, analytic and geometric aspects of singularities, particularly in solutions of longstanding problems posed by Whitney, Thom and Hironaka - the originators of the singularity theories in geometry and algebra. This also led me to an extension of classical Sobolev-Nirenberg and Bernstein-Markov inequalities to a singular setting, and to a discovery of `tame' subanalytic sets on which one can do classical local analysis.

奇点以类似于日常语言的方式表达了许多数学分支中形式的不规则性，并且是大多数数学及其应用的基本研究对象。形式的重要特征往往集中在奇点上。我研究的目的是找到奇异对象的几何编码信息与奇异对象分析之间的联系。这使我发现了奇异点的代数、解析和几何方面之间的基本联系，特别是在解决几何和代数奇点理论的创始人惠特尼、汤姆和弘中提出的长期问题方面。这也使我将经典的索博列夫-尼伦堡和伯恩斯坦-马尔可夫不等式扩展到奇异的环境，并发现了“驯服”的亚分析集，人们可以在其上进行经典的局部分析。