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).

奇点在数学的许多分支中以一种类似于它在日常语言中的含义的方式表达形式的不规则性，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。我的研究目标是找到编码在奇异物体的几何和分析的信息之间的联系。这使我发现了代数，分析和几何方面的奇点之间的基本联系，特别是在解决惠特尼，托姆和Hironaka提出的长期问题-几何和代数奇点理论的创始人。这也使我扩展到经典的Sobolev，尼伦贝格和伯恩斯坦，马尔可夫不等式的奇异设置，并发现了'驯服' subanalytic集上，人们可以做经典的本地分析。在过去的几年里，我的工作导致了A。通过去奇异化的方法构造远离奇异点的完整庞加莱型卡勒度量;B.在大于一的维度上发现欧几里德除法;C.发现并建设性地描述了“普遍分层”; D.一个分类的所有'最小奇点'的Kollar为三倍。当然，在所有这些不同的问题上还有许多工作要做。除了我的研究，我也很幸运，有三个学生毕业，在过去的两年里，优秀的博士学位。论文我肯定会继续工作的问题上面列出的，但我最希望继续产生优秀的数学家使用肥沃的土壤的多样性，这些问题。最后，在2011年和2012年，我证明了两个著名的长期假设：1。Hironaka在1977年提出的关于奇异点分解的Q-普适性，以及2. Vasconcelos猜想的几何特征的平坦性（完全揭秘这个代数概念）。