Desingularization and applications. Analysis on and Geometry of singular spaces

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异性的代数、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;扩展了经典Bernstein-Markov和Gagliardo-Nirenberg型分析不等式的奇异性设置;尖锐的新的边界上的热核viadesingularizing重量，切马尔可夫不等式的奇异品种，利用Chow型定理和去奇异化的方法，构造了一个关于奇点的Poincare型度量。在过去的6年里，我发现了一个关于奇点的Poincare型度量的简单构造。Bertini型定理对我描述满足Thom和Whitney-a条件的泛分层至关重要，它建立了经典去奇异化的复杂性界限（结果是非常高的）和低的复杂性界限（多项式）对于基本维度2中的归一化纳什去奇异化，证明了几何Auslander标准的开放性和平坦性的代数态射，也推进了我15岁的几何最小模型的程序分类的“最小奇点”在4和3个变量（即一个最小的奇异性列表，除了正常的交叉与存在的desingularizationsisomorphic关闭这些奇点，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（在任何维度上，它等价于诱导度量的去奇异化），并将我的弧解析性和Malgrange型除以解析函数的结果推广到拟解析类。我计划对这些结果进行自然扩展，主要目标是在去奇异化和奇异空间的几何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复杂的射影代数流形扩展到代数子簇的相同维度，并在一个封闭的集上的功能，这是一个单独的限制半代数函数和一个k次可微函数是一个限制的k次可微半代数函数，等等。我试图澄清代数、几何和分析中的各种问题，这被证明是培养研究生中优秀数学家的坚实基础。我的研究计划是重复这种经历。

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.