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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758689
• 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 andits applications are central to my work.In the past years my research led me to discovery of fundamental links between algebraic, analyticand 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 andGagliardo-Nirenberg type inequalities of analysis; to sharp new bounds on the heat kernel viadesingularizing weights, to tangential Markov inequalities on singular varieties, to a Chow typetheorem for ideals and by means of the latter and desingularization to a simple construction of aPoincare type metric off singularities.Within the last 6 years I discovered a Bertini-type theorem crucial for my characterization ofUniversal Stratifications satisfying Thom and Whitney-a conditions, established complexity boundsfor classical desingularization (turned out to be very high) and low complexity bounds (polynomial)for Normalized Nash Desingularization in essential dimension 2 , proved Geometric Auslandercriteria 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 desingularizationsisomorphic off these singularities, e.g. just the Whitney Umbrella for surfaces). Recently I alsoestablished 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 andMalgrange 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 theinformation 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-timesdifferentiable 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 afertile ground for raising excellent mathematicians from graduate students. My research plansaim to repeat this experience.

奇点在数学及其应用的许多分支中表达了形式的不规则性。形式的重要特征往往集中在奇点处。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：惠特尼、汤姆和弘中提出的长期存在的问题的解决方案；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，到奇异簇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奇点之外的庞加莱型度量的简单构造。在过去的 6 年里，我发现了一个 Bertini 型定理，它对于我满足 Thom 和 Whitney-a 条件，为基本维度 2 中的归一化纳什去奇异化建立了经典去奇异化的复杂性界限（结果非常高）和低复杂性界限（多项式），证明了代数态射的开放性和平坦性的几何 Auslander 准则，并将我 15 年的“几何最小模型”程序推进到了 4 和 4 中“最小奇点”的分类 3 个变量（即除了正常交叉之外的最小奇点列表，并且存在这些奇点的去奇异化同构，例如表面的惠特尼伞）。最近，我还建立了奇异三重余切丛的去奇异化（并且，在任何维度，它等价于诱导度量的去奇异化），并将我的奥解析性和马尔格朗日类型除法通过解析函数结果扩展到准解析类。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编码和分析中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品种，并且闭集上的函数分别是半代数函数和k次可微函数的限制是k次可微半代数函数的限制，等等。最后，我的尝试 阐明代数、几何和分析中的各种问题被证明是从研究生中培养优秀数学家的沃土。我的研究计划旨在重复这一经历。