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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635459
• 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, and2. Vasconcelos conjecture on a geometric characterization of flatness (fully demystifying this algebraic notion).

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。我的研究目的是找到奇异物体的几何编码信息与分析之间的联系。这使我发现了奇异性的代数、解析和几何方面之间的基本联系，特别是在解决长期存在的问题时惠特尼、汤姆和弘中——几何和代数奇点理论的创始人。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。在过去的几年里，我的工作成果是 A. 通过去奇异化构建了一个完整的庞加莱型卡勒度量。发现维数大于一的欧几里得除法；C. “普遍分层”的发现和建设性描述；D． Kollar 的所有“最小奇点”的三重分类。当然，解决所有这些不同的问题还有很多工作要做。除了我的研究之外，我还很幸运地拥有三名学生在过去两年中以优异的博士学位毕业。论文。我肯定会继续研究上面列出的问题，但我最希望的是利用这些问题多样性的沃土继续培养优秀的数学家。最后，我在2011年和2012年证明了两个众所周知的长期猜想：1。 Hironaka 于 1977 年提出了奇点消解的 Q 通用性，以及 2。瓦斯康塞洛斯关于平坦度几何特征的猜想（完全揭开了这个代数概念的神秘面纱）。