Desingularization and applications. Analysis on and Geometry of singular spaces

奇点在许多数学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。去世俗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、分析和几何方面之间的基本联系：解决了Whitney、Thom和Hironaka提出的长期问题；扩展到经典的Bernstein-Markov和gagliardo - nirenberg型不等式的奇异设置；讨论了热核的奇异化权值上的尖锐的新界，讨论了奇异变异体上的切向马尔可夫不等式，讨论了理想的Chow型定理，并利用后者和去物化，讨论了离奇异点的aPoincare型度量的一个简单构造。在过去的6年里，我发现了一个bertini型定理，这对我描述满足Thom和Whitney-a条件的普遍分层至关重要，建立了经典非广域化的复杂性边界（后来证明是非常高的）和基本维2的规范化纳什非广域化的低复杂性边界（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4和3个变量的“最小奇点”分类。除了正规交叉点之外的最小奇异点列表，这些奇异点存在去个体性同纯性，例如曲面的惠特尼伞)。最近，我还建立了奇异三折的余切束的非广域化（并且，在任何维度上，它等价于诱导度规的非广域化），并将解析函数的不可解析性和malgrange型划分结果推广到拟解析类。我计划研究这些结果的自然延伸。本研究的主要目的是在奇异空间的几何和分析中发现非物化与可能编码的信息之间的更紧密联系。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 - small复合体的复解析稳定叶扩展到相同维数的代数子变种，并且该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，我试图澄清代数、几何和分析方面的各种问题，这被证明是培养研究生优秀数学家的沃土。我的研究计划重复这一经历。

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.