Desingularization and applications. Analysis on and Geometry of singular spaces

去奇异化和应用。

• 批准号: RGPIN-2018-04445
• 负责人: Milman, Pierre
• 金额: $ 1.46万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677118
• 关键词: Desingularization; applications.; Analysis; Geometry; singular

Singularities express irregularities of form in many branches of mathematics and its applications.*The important features of forms are often concentrated at singularities. Desingularization and*its applications are central to my work.*******In the past years my research led me to discovery of fundamental links between algebraic, analytic*and geometric aspects of singularities: solutions of the long-standing problems posed by Whitney,*Thom and Hironaka; to an extension to a singular setting of the classical Bernstein-Markov and*Gagliardo-Nirenberg type inequalities of analysis; to sharp new bounds on the heat kernel via*desingularizing weights, to tangential Markov inequalities on singular varieties, to a Chow type*theorem for ideals and by means of the latter and desingularization to a simple construction of a*Poincare type metric off singularities.*******Within the last 6 years I discovered a Bertini-type theorem crucial for my characterization of*Universal Stratifications satisfying Thom and Whitney-a conditions, established complexity bounds*for classical desingularization (turned out to be very high) and low complexity bounds (polynomial)*for Normalized Nash Desingularization in essential dimension 2 , proved Geometric Auslander*criteria for openness and for flatness of algebraic morphisms and also advanced my 15 years old*`geometric minimal models' program to a classification of `minimal singularities' in 4 and 3 variables*(i.e. a minimal list of singularities besides normal crossings with existence of desingularizations*isomorphic off these singularities, e.g. just the Whitney Umbrella for surfaces). Recently I also*established desingularization of the cotangent bundle of singular threefolds (and, in any dimension,*its equivalence to a desingularization of the induced metric) and extended my arcanalyticity and****Malgrange type division by analytic functions results to the quasi-analytic classes. I plan to work on natural extensions of these results.******The main objective of the proposed research is to find a closer link between desingularization and the*information that may be encoded in the geometry of and analysis on singular spaces. My longer term objective would be a reconstruction of the entire process of resolution of singularities in terms of geometry and/or analysis. I also would like to show that complex analytic stable leaves of a `rigid' Morse-Smale complexes of projective algebraic manifolds extend to algebraic subvarieties of the same dimension, and that function on a closed set which is separately a restriction of a semi-algebraic function and of a k-times*differentiable function is a restriction of a k-times differentiable semi-algebraic function, etc. ******Finally, my attempts to clarify diverse problems in algebra, geometry and analysis proved to be a*fertile ground for raising excellent mathematicians from graduate students. My research plans*aim to repeat this experience.***

奇异性在数学及其应用的许多分支中表达形式的不规则性。*形式的重要特征通常集中在奇点上。 *****在过去的几年中*******，我的研究使我发现了代数，分析*与奇异之处的几何方面：惠特尼（Whitney），*汤姆（Thom）和hironaka所构成的长期问题的解决方案；扩展到经典的伯恩斯坦 - 马尔科夫（Bernstein-Markov）和*gagliardo-nirenberg类型的分析不平等现象；通过*降低的重量在热内核上的新界限，以奇异品种的切向马尔可夫的不平等，对理想和后者和后者的理论*理论的简单构想，从而简单地构造了一个*poincare类型的度量，我在过去的6年中发现了我的特征。 Thom和Whitney-A条件，为经典的垂缘化（原来是很高）和低复杂性界限（多项式）*的确定复杂性界限*，用于基本维度2中的nash降低的标准化，几何的几何auslander*标准是开放性和对代数形式的纯度和对我的15年的平稳性的标准。 4和3变量中的“最小奇异性”*（即，除了正常的穿越外，奇异性列表的最小清单与存在降低的奇异性*同构相反，例如只是惠特尼伞的表面）。最近，我还*建立了奇异三倍的cotangent束（并且在任何维度上*它等于诱导的度量标准），并通过分析函数结果扩展了我的Arcanalytictions和**** Malgrange类型分裂的结果。我计划研究这些结果的自然扩展。******拟议的研究的主要目的是在降低和*在奇异空间的几何形状和分析中可能编码的*信息之间找到更紧密的联系。我的长期目标将是对几何和/或分析的整个奇异性解决过程的重建。 I also would like to show that complex analytic stable leaves of a `rigid' Morse-Smale complexes of projective algebraic manifolds extend to algebraic subvarieties of the same dimension, and that function on a closed set which is separately a restriction of a semi-algebraic function and of a k-times*differentiable function is a restriction of a k-times differentiable semi-algebraic function, etc. ******Finally, my事实证明，试图阐明代数，几何学和分析的各种问题是*肥沃的基础，可以从研究生中培养出色的数学家。我的研究计划*旨在重复这一经验。***