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: 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=568632
• 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. In the several last years my work resulted in A. a construction of a complete Poincare type Kahler metric off singularities by means of desingularization; B. a discovery of an Euclidean division in dimension larger than one; C. a discovery and a constructive characterization of `universal stratifications'; D. a classification of all `minimal singularities' of Kollar for threefolds. Of course a lot is still left to be done in all these diverse problems. Besides my research I was also fortunate to have three students graduating in the last two years with excellent Ph.D. theses. I surely will continue to work on the problems listed above, however I most of all hope to continue to produce excellent mathematicians using the fertile ground of the diversity of these problems. Finally, in 2011 and 2012 I have proved two well-known longstanding conjectures: 1. posed by Hironaka in 1977 on the Q-universality of resolution of singularities, and 2. Vasconcelos conjecture on a geometric characterization of flatness (fully demystifying this algebraic notion).

奇点在许多数学分支中表达形式的不规则性，其方式类似于它在日常语言中的含义，并且是大多数数学及其应用中的基本研究对象。形式的重要特征往往集中在奇点上。我的研究目的是找到奇点的几何学和分析中所编码的信息之间的联系。这使我发现了奇点的代数、解析和几何方面之间的基本联系，特别是在解决惠特尼、托姆和平中--几何和代数奇点理论的创始人--提出的长期存在的问题时。这也使我将经典的Sobolev-Nirenberg和Bernstein-马尔可夫不等式推广到奇异环境，并发现了可以在其上进行经典局部分析的“驯服”次分析集。 在过去的几年里，我的工作导致了 A.用去奇异的方法构造了完备的Poincare型Kahler度量； B.发现维度大于1的欧几里得除法； C.对“普遍分层”的发现和建设性描述； D.关于三重Kollar的所有‘极小奇点’的分类。 当然，在所有这些不同的问题上，还有很多事情要做。除了我的研究，我也很幸运，在过去的两年里，有三个学生以出色的博士论文毕业。我肯定会继续研究上面列出的问题，但我最希望的是继续培养出优秀的数学家，利用这些问题的多样性这一沃土。 最后，在2011年和2012年，我证明了两个众所周知的长期存在的猜想： 1.由Hironaka在1977年提出的关于奇点分解的q-普适性，以及 2.关于平坦度的几何刻画的瓦斯康塞洛斯猜想(完全揭开了这一代数概念的神秘面纱)。