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.***

奇点表达了数学及其应用的许多分支中形式的不规则性。*形式的重要特征通常集中在奇点处。去奇异化及其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奇点的代数、解析*和几何方面之间的基本联系：解决长期存在的问题作者：惠特尼、*汤姆和弘中；扩展到经典伯恩斯坦-马尔可夫和*加利亚多-尼伦伯格型分析不等式的奇异设置；通过*去奇异化权重，到热核上尖锐的新界限，到奇异品种上的切向马尔可夫不等式，到理想的Chow型*定理，并通过后者和去奇异化到a * Poincare型度量的简单构造。 ******在过去的六年里，我发现了一个贝尔蒂尼型定理，它对于我描述*满足汤姆和惠特尼条件的万有分层至关重要，建立了经典去奇异化的复杂性界限*（事实证明非常高）和低复杂性界限（多项式）*基本维度 2 中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标准，并且还推进了我的 15岁*“几何最小模型”程序，用于对 4 和 3 个变量中的“最小奇点”进行分类*（即最小列表除了正常交叉之外的奇点，并且存在这些奇点的去奇异化*同构，例如只是表面的惠特尼伞）。最近，我还*建立了奇异三重的余切丛的去奇异化（并且，在任何维度上，*它等价于导出度量的去奇异化），并将我的神秘分析性和****通过解析函数结果的马尔格朗日类型除法扩展到了准- 分析课程。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中编码和分析的*信息之间更紧密的联系。我的长期目标是在几何和/或分析方面重建解决奇点的整个过程。我还想证明射影代数流形的“刚性”Morse-Smale 复形的复解析稳定叶扩展到相同维度的代数子族，并且该函数在闭集上，该闭集分别是半代数的限制函数和 k 次*可微函数是 k 次可微半代数函数的限制，等等。 ******最后，我尝试澄清各种问题在代数、几何和分析领域，事实证明是从研究生中培养出优秀数学家的肥沃土壤。我的研究计划*旨在重复这一经历。***