Geometry and analysis on singular spaces

奇异空间的几何与分析

• 批准号: RGPIN-2017-06537
• 负责人: Bierstone, Edward
• 金额: $ 2.7万
• 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=710728
• 关键词: Geometry; analysis; singular; spaces

The proposal is in the general areas of algebraic or analytic geometry, real analysis, and relations between them. The overall objective is a better understanding of singularities, which describe irregularities of spaces and functions in all branches of mathematics and its applications. The proposal focuses on resolution of singularities and applications of desingularization techniques, of an algorithmic or differential nature, to questions in subanalytic geometry and classical real analysis. Several problems of resolution of singularities of great current interest (e.g., partial desingularization, desingularization of vector fields and differential forms, monomialization of mappings, desingularization in positive characteristic) have seen remarkable recent progress in dimensions up to three; further progress depends on overcoming obstacles that have certain common features. The proposal is organized in terms of five general projects. Progress should shed light on long-standing conjectures and open new directions of research. The main subjects are: (1) Partial resolution of singularities. The goal is to find models of birational equivalence classes with mild singularities that have to be admitted in natural situations. (For example, to simultaneously resolve the singularities of a family of curves, one has to allow special fibres with transverse self-intersections.) (2) Desingularization of differential forms, with a view towards monomialization of a metric and applications to L2 cohomology. (3) Equisingularity problems related to the invariants which determine an algorithm for resolving singularities. (4) Subanalytic geometry. Semialgebraic and more general subanalytic sets are ubiquitous in many areas of mathematics; the goal is a global smoothing theorem for subanalytic sets, capturing the birational or bimeromorphic feature of resolution of singularities. (5) Singularities in real analysis--problems on the common border of algebraic geometry and analysis, concerning quasianalytic functions (of interest in both partial differential equations and model theory), and Fefferman-Whitney extension of functions defined on closed sets, preserving geometric classes.

该提案涉及代数或解析几何、实分析以及它们之间的关系的一般领域。总体目标是更好地理解奇点，奇点描述了数学及其应用所有分支中空间和函数的不规则性。该提案的重点是奇点的解决以及算法或微分性质的去奇异化技术的应用，以解决亚解析几何和经典实分析中的问题。当前备受关注的奇点解决的几个问题（例如，部分去奇异化、向量场和微分形式的去奇异化、映射的单项化、正特征的去奇异化）最近在高达三维的领域取得了显着的进展；进一步的进展取决于克服具有某些共同特征的障碍。 该提案由五个一般项目组成。进展应该会阐明长期存在的猜想并开辟新的研究方向。主要课题有：（1）奇点的部分解决。目标是找到在自然情况下必须承认的具有温和奇点的双有理等价类模型。 （例如，为了同时解决一系列曲线的奇异性，必须允许具有横向自相交的特殊纤维。）（2）微分形式的去奇异化，着眼于度量的单项化及其在 L2 上同调中的应用。 (3) 与决定解决奇点算法的不变量相关的等奇点问题。 (4)亚解析几何。半代数和更一般的亚解析集在数学的许多领域中普遍存在。目标是亚解析集的全局平滑定理，捕获奇点解析的双有理或双亚纯特征。 (5)实分析中的奇点——代数几何和分析的共同边界上的问题，涉及准解析函数（偏微分方程和模型理论中感兴趣的），以及在闭集上定义的函数的费弗曼-惠特尼扩展，保留几何类。