Desingularization and applications. Analysis on and Geometry of singular spaces

在数学及其应用的许多分支中，奇点表示形式的不规则性。形式的重要特征往往集中在奇点。去单角化及其应用是我工作的中心。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、*托姆和平中提出的长期问题的解决方案；扩展到经典的伯恩斯坦-马尔可夫和*Gagliardo-Nirenberg型分析的奇异设置；在过去的6年里，我发现了一个贝尔蒂尼型定理，它对刻画满足Thom和Whitney-a条件的普适分层至关重要，为经典的去奇异化建立了复杂性(证明是非常高的)和低复杂性(多项式)的界。证明了代数态射的开放性和平坦性的几何Auslander*标准，并将我已有15年历史的*“几何极小模型”程序改进为4和3个变量的“最小奇点”分类*(即存在与这些奇点同构的正常交叉点之外的最小奇点列表，例如曲面的惠特尼伞)。最近，我还建立了奇异三重余切丛的去奇化(并且，在任何维度上，它等价于诱导度量的去奇化)，并将我的弧解析性和*Malgrange型按解析函数除法的结果推广到拟解析类。我计划致力于这些结果的自然扩展。*拟议研究的主要目标是在去单元化和可能编码在奇异空间的几何和分析中的*信息之间找到更紧密的联系。我的长期目标是从几何和/或分析的角度重建奇点分解的整个过程。此外，我还想证明射影代数流形的“刚性”Morse-Smer复形的复解析稳定叶扩张到相同维度的代数子簇，闭集上的函数分别是半代数函数的限制和k次可微函数的限制，等等。*最后，我试图澄清代数、几何和分析中的各种问题，证明这是培养研究生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.***