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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712971
• 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.

奇点在数学的许多分支及其应用中表示形式的不规则性。 形式的重要特征往往集中在奇点上。去舌化和 它的应用是我工作的核心。 在过去的几年里，我的研究使我发现了代数，分析和数学之间的基本联系。 奇异点的几何方面：惠特尼提出的长期问题的解决方案， Thom和Hironaka;对经典Bernstein-Markov的奇异设置的扩展， 分析的Gagliardo-Nirenberg型不等式;通过构造新的热核界 去奇异化权，到奇异簇上的切马尔可夫不等式，到Chow型 定理的理想，并通过后者和desingularization到一个简单的建设， 奇点下的Poincare型度量。 在过去的6年里，我发现了一个贝尔蒂尼型定理，这对我描述 满足Thom和Whitney-a条件的泛分层，建立了复杂性界 对于经典的去奇异化（结果是非常高的）和低复杂度界限（多项式） 对于本质维数为2的归一化纳什去奇异化，证明了几何Auslander 代数态射的开放性和平坦性的标准， “几何最小模型”程序，对4和3个变量的“最小奇点”进行分类 (i.e.存在去奇异化的除正常交叉外的最小奇点列表 同构的这些奇点，例如，只是惠特尼伞的表面）。最近我也 建立了奇异三重余切丛的去奇异化（并且，在任何维度上， 它等价于诱导度量的去奇异化），并扩展了我的弧解析性， Malgrange型除解析函数的结果准解析类。我计划研究这些结果的自然扩展。 拟议研究的主要目标是找到去奇异化和 可以在奇异空间的几何结构和奇异空间上的分析中编码的信息。我的长期目标将是在几何和/或分析方面重建奇点的整个解决过程。我还想表明，复杂的分析稳定叶的一个'刚性' Morse-Smale复合物的射影代数流形延伸到代数子簇的相同的层面，并在一个封闭的功能集，这是一个半代数函数和一个k倍的限制 可微函数是k次可微半代数函数的限制等。 最后，我试图澄清代数，几何和分析中的各种问题，证明是一个很好的方法。 培养优秀数学家的沃土。我的研究计划 我们的目标是重复这种经验。