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.

我们将从几何和解析的角度研究奇异半代数和次解析集。在几何方面，我们将研究这些集合的内部度量性质，特别是在奇异轨迹处和附近的微分几何量的行为：测地线，局部体积增长函数，内在距离函数。在分析方面，我们将专注于调和形式，热核渐近和L2霍奇理论的正则性问题。这是密切相关的几何问题，因为正确理解的距离函数是中央研究的预解式的拉普拉斯-贝尔特拉米算子。我们的方法的关键点是使用的决议的奇异性，适应度量结构，并适应退化的度量等决议的pseudodifferential微积分的发展。