Geometry and analysis of semi-algebraic sets

半代数集的几何与分析

• 批准号: 5453760
• 负责人: Professor Dr. Daniel Grieser
• 金额: --
• 依托单位: Institut für Mathematik (IFM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5453760?language=en
• 关键词: Geometry; analysis; semi; algebraic; sets

We will study singular semi-algebraic and sub-analytic sets from a geometric and an analytic point of view. On the geometric side we will study the inner metric properties of such sets, and in particular the behavior of differential geometric quantities at and near the singular locus: geodesies, the local volume growth function, the intrinsic distance function. On the analytic side we will focus on regularity questions for harmonic forms, heat kernel asymptotics and L2 Hodge theory. This is closely related to the geometric questions since a proper understanding of the distance function is central to the study of the resolvent of the Laplace-Beltrami operator. Key points of our approach are the use of resolutions of singularities, adapted to the metric structure, and the development of a pseudodifferential calculus adapted to the degenerations of the metric on such resolutions.

我们将从几何学和分析的角度研究奇异的半分析集和亚分析集。在几何方面，我们将研究此类集合的内部度量特性，尤其是在奇异基因座和附近差异几何量的行为：地理位置：局部体积生长函数，固有距离函数。在分析方面，我们将重点介绍谐波形式，热核渐近学和L2霍奇理论的规律性问题。这与几何问题密切相关，因为对距离函数的正确理解对于拉普拉斯 - 贝特拉米操作员的分解的研究至关重要。我们方法的要点是使用奇异性的分辨率，适合于度量结构，以及在此类分辨率上适用于指标退化的假差分计算的发展。