Desingularization and applications. Analysis on and Geometry of singular spaces

去奇异化和应用。

• 批准号: RGPIN-2018-04445
• 负责人: Milman, Pierre
• 金额: $ 1.46万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712971
• 关键词: Desingularization; applications.; Analysis; Geometry; singular

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.

奇点在许多数学分支及其应用中表示形式的不规则性。