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点的代数、分析 * 和几何方面之间的基本联系：解决了惠特尼、* 汤姆和Hironaka提出的长期问题;扩展了经典伯恩斯坦-马尔可夫和 *Gagliardo-Nirenberg型分析不等式的奇异设置;通过去奇异化权重，在热核上建立尖锐的新边界，在奇异簇上建立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的 *Poincare型度量的简单构造。*在过去的6年里，我发现了一个贝尔蒂尼型定理，它对我描述满足Thom和Whitney-a条件的泛分层至关重要，为经典的去奇异化建立了复杂性界限（结果是非常高的）和低的复杂性界限（多项式）* 对于本质维度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，也先进了我15岁 * 的“几何最小模型”程序的分类“在4和3个变量 * 中的最小奇异点（即除了正常交叉之外的奇异点的最小列表，存在与这些奇异点同构的去奇异化 *，例如，对于曲面只有惠特尼伞）。最近我还 * 建立了去奇异化的余切丛的奇异三倍（和，在任何方面，* 其等价于一个去奇异化的诱导度量），并延长我的arcanalyticity和 *Malgrange型除法解析函数的结果准解析类。我计划对这些结果进行自然扩展。*拟议的研究的主要目标是找到一个更紧密的联系之间的desingularization和 * 的信息，可能被编码在几何和分析奇异空间。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簇，并且分别是半代数函数和k次可微函数的限制的闭集上的函数是k次可微半代数函数的限制，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 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.***