Geometry and analysis of semi-algebraic sets

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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.