Geometry and analysis on singular spaces

该建议是在一般领域的代数或解析几何，真实的分析，以及它们之间的关系。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的空间和函数的不规则性。该建议的重点是解决奇异性和应用的desingularization技术，算法或微分性质，在次解析几何和经典的真实的分析的问题。目前人们非常感兴趣的几个奇异点的解决问题（例如，部分去奇异化，向量场和微分形式的去奇异化，映射的单单性化，正特征的去奇异化）在高达3维的维度中已经看到了显着的最新进展;进一步的进展取决于克服具有某些共同特征的障碍。该提案分为五个一般项目。这些进展应能阐明长期存在的问题，并开辟新的研究方向。主要内容有：（1）奇点的部分分解。我们的目标是找到模型的双有理等价类具有温和的奇异性，必须承认在自然情况下。(For例如，为了同时解决一族曲线的奇点，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，着眼于度量的单值化和对L2上同调的应用。(3)等奇异性问题与确定解决奇异性的算法的不变量有关。(4)次解析几何半代数和更一般的次解析集在数学的许多领域都是普遍存在的;目标是次解析集的全局光滑定理，捕获奇异性分解的双有理或双半纯特征。(5)奇异性在真实的分析-问题的共同边界的代数几何和分析，有关准解析函数（感兴趣的偏微分方程和模型理论），和February man-Whitney扩展的功能定义在封闭的集合，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.