Geometry and analysis on singular spaces

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Singularities describe irregularities of spaces and of functions in many branches of mathematics and its applications. The general goal of our recent research and this proposal is to better understand singularities in birational geometry, analysis and subanalytic geometry. These subjects span a vast area. The questions depend on the classes of functions considered (algebraic, analytic, differentiable, formal). Our previous work has developed "differential calculus" techniques for singular spaces, that apply simultaneously to the various classes, leading to the discovery of links among algebraic, geometric, combinatorial and analytic aspects of singularities (for example, the surprising role of differentiable functions in subanalytic geometry). Our algorithm for constructive resolution of singularities has opened up a previously inaccessible area.