Geometry and analysis on singular spaces

该提案涉及代数或解析几何、实分析以及它们之间的关系等一般领域。总体目标是更好地理解奇点，奇点描述了数学及其应用的所有分支中空间和函数的不规则性。该建议侧重于解决奇点和去奇点技术的应用，算法或微分性质的问题，在次解析几何和经典的实数分析中。解决当前非常感兴趣的奇点的几个问题(例如，部分去奇点化、矢量场和微分形式的去奇点化、映射的单项化、正特征中的去奇点化)最近在三个维度上取得了显著的进展；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。这一进展应该会揭示长期存在的猜想，并开辟新的研究方向。主要内容有：(1)奇点的部分分解。我们的目标是找到在自然情况下必须承认的具有轻微奇点的种族等价类的模型。(例如，为了同时解决曲线族的奇点，人们必须允许具有横向自交的特殊纤维。)(2)微分形式的去奇点化，以期实现度量的单项化并应用于L2上同调。(3)与决定奇点分解算法的不变量有关的均衡性问题。(4)次解析几何。半代数集和更一般的次分析集在许多数学领域中普遍存在；目标是建立次分析集的全局光滑化定理，以捕捉奇点分解的双整体或双纯特征。(5)实分析中的奇点--代数几何和分析的共同边界上的问题，涉及拟解析函数(偏微分方程式和模型理论都感兴趣)，以及定义在闭集上的函数的Fefferman-Whitney扩张，保持几何类。

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.