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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=686938
• 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.

该提案在代数或分析几何形状，真实分析及其之间的关系的一般领域。总体目标是更好地理解奇点，它描述了数学及其应用的所有分支中空间和功能的不规则性。该提案的重点是解决奇异技术的分辨率，算法或差异性质的降低技术的应用，介绍了亚分析几何形状和经典真实分析中的问题。分辨出巨大兴趣的奇异性的几个问题（例如，部分降低，向量场的降低形式和差异形式，映射的单次化，呈阳性特征中的降落化）已在多达三个方面取得了显着的进展。进一步的进步取决于克服具有某些共同特征的障碍。******该提案是根据五个通用项目组织的。进步应阐明长期的猜想和开放的新研究方向。主要主题是：（1）奇异性的部分分辨率。目的是找到具有轻度奇异性的男性等价类别的模型，这些模型必须在自然情况下接受。 （例如，为了同时解决曲线家族的奇异性，必须允许具有横向自相互作用的特殊纤维。）（2）差异形式的降低，以观察到L2 cohomology的度量和应用。 （3）与不变性有关的方程性问题，这些问题确定了解决奇异性算法的算法。 （4）亚分析几何形状。在许多数学领域，半ge和更一般的亚分析集无处不在。该目标是用于亚分析集的全球平滑定理，捕获了分辨率分辨率的生育或双形晶状体特征。 （5）真实分析中的奇异性 - 关于代数几何和分析的共同边界的问题，涉及准二元分析函数（在部分微分方程和模型理论中都具有意义），以及Fefferman-Whitney-whitney-whitney在封闭集合上定义的功能扩展，并保留了几何学类别。