Geometry and analysis on singular spaces

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Singularities in mathematics describe irregularities of spaces and functions, in the same way that a singularity in everyday language means an irregularity of an object or of behaviour. Singularity is thus a basic idea in most mathematical disciplines and their applications. The important features of an object are often concentrated at the singularities. The goal of my research is to better understand singularities using the powerful tools of several areas of mathematics -- algebra, analysis and geometry.Many important questions in mathematics depend on the class of spaces or functions considered (algebraic, analytic, differentiable,...). I have tried to take a universal approach, developing a "differential calculus" for singular spaces, that applies simultaneously to the various classes. This has led to the discovery of unexpected links among algebraic, combinatorial and analytic aspects of singularities, and to the solution of problems of Whitney, Thom and Hironaka -- the pioneers of singularity theory.My relatively elementary algorithm for resolution of singularities (projection onto a singular object of a space that is smooth, or nonsingular) helps to unravel the hidden features of singular spaces. It opens a previously inaccessible area, and provides techniques to attack fundamental problems in diverse areas like logic and analysis.The problems that I am working on are potentially important in the physical and information sciences. Efficient implementation of algorithms for resolution of singularities, for example, is an active area of research in computer algebra.