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.

奇异性在数学的许多分支中表达了形式的不规则性，其方式与日常语言的含义相似，并且是大多数数学及其应用中研究的基本对象。形式的重要特征通常集中在奇点上。我的研究的目的是找到在几何形状中编码的信息与对奇异对象的分析之间的联系。这使我发现了奇异性的代数，分析和几何方面之间的基本联系，尤其是在惠特尼，托姆和霍里亚纳克（Hironaka）长期存在的问题的解决方案中 - 奇异和alirake singulory singulor the veomeories in Geomelories of Geolabry of Geomely of veomry of veomry of veomry in velbrry in velbrry in velbrry and veolabry and ecormy and ecombry and ecormy and ealbrry sissecor。这也使我延伸了古典的Sobolev-Nirenberg和Bernstein-Markov的不平等现象，并发现了“驯服”亚分析集，可以在其中进行经典的本地分析。