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。在过去的几年中，我的工作导致A.通过降低的构造构造了一个完整的庞美列型卡勒度量标准。b。在大于一个; c的维度中发现欧几里得分裂的发现。 “通用分层”的发现和建设性表征； d。三倍的Kollar的所有“最小奇点”的分类。当然，在所有这些不同的问题中，仍然有很多事情要做。除了我的研究外，我还很幸运能在过去两年中获得三名学生，并获得出色的博士学位。这些。我肯定会继续处理上面列出的问题，但是我最重要的是希望继续使用这些问题的多样性来生产优秀的数学家。最后，在2011年和2012年，我证明了两个众所周知的长期猜想：1。 Hironaka于1977年就奇异性解决方案和2。 Vasconcelos对平坦度的几何表征的猜想（完全揭开了这个代数概念）。