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异点上。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、解析*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型不等式的奇异集；通过权值的去具体化，得到热核的新边界，得到奇异变异上的切向马尔可夫不等式，得到理想的Chow型定理，并利用后者和去具体化得到奇异点上庞加莱型度规的简单构造。*******在过去的6年里，我发现了一个bertini型定理，对于我表征满足Thom和Whitney-a条件的普遍分层至关重要，建立了经典去广域化的复杂性界限*（结果非常高）和基本维2的规范化纳什去广域化的低复杂性界限（多项式）*。证明了代数态射的开放性和平坦性的几何Auslander标准，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奇异点的最小列表，这些奇异点存在去象素化*同构，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），并将我的不可解析性和****Malgrange型划分的解析函数结果推广到拟解析类。我计划研究这些结果的自然延伸。******所提出的研究的主要目标是在去物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”莫尔斯-小复合体的复解析稳定叶扩展到相同维数的代数子变量，并且该函数在封闭集上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，我试图阐明代数中的各种问题，事实证明，几何和分析是培养研究生优秀数学家的沃土。我的研究计划旨在重复这一经历

Singularities express irregularities of form in many branches of mathematics and its applications.*The important features of forms are often concentrated at singularities. Desingularization 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.***