Geometry and analysis of semi-algebraic sets

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，和发展的pseudodifferential适应退化的度量等决议。

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.