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上。去单一化和 它的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代数、分析 奇点的几何方面：惠特尼提出的长期问题的解决方案， Thom和Hironaka；推广到经典Bernstein-Markov型和 分析的Gagliardo-Nirenberg型不等式；通过 将奇异簇上的切向马氏不等式的权重去单一化为Chow型 关于理想的一个定理，并借助于后者和去奇异化为一个简单的构造 Poincare类型度规的奇点。 在过去的6年里，我发现了一个贝尔蒂尼型定理，这对我刻画 满足Thom和Whitney-a条件的普适分层，建立了复杂性界 对于经典去奇异化(结果是非常高的)和低复杂性界(多项式) 对于本质维为2的归一化Nash去单值化，证明了几何Auslander 代数态射的开性和平坦性的判据，也比我15岁早 “几何极小模型”程序对4个和3个变量的“最小奇点”的分类 (即，除正常交叉点外存在去单角化的奇点的最小列表 与这些奇点同构，例如仅曲面的惠特尼伞)。最近我也 奇数三重余切丛的已建立的去单角化(并且，在任何维度中， 它等价于诱导度量的去奇异)，并扩展了我的弧解析性和 用解析函数进行Malgrange型除法得到拟解析类。我计划致力于这些结果的自然延伸。 这项研究的主要目的是在去单语化和语言学习之间找到一个更紧密的联系 可以在奇异空间的几何和分析中编码的信息。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。我还想证明，射影代数流形上的“刚性”Morse-Smer复形的复解析稳定叶扩张到相同维代数子簇，以及闭集上的函数分别是半代数函数和k次函数的限制 可微函数是k次可微半代数函数的限制等。 最后，我试图澄清代数、几何和分析中的各种问题，结果证明是一种 从研究生中培养优秀数学家的沃土。我的研究计划 目的是重演这段经历。

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.