Geometry and analysis on singular spaces

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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.