Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.

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Singularities express irregularities of form in many branches of mathematics in a way similar to what it means in an everyday language and are a basic object of study in most of the mathematics and its applications. The important features of form are often concentrated at singularities. The objective of my research is to find links between the information encoded in the geometry of and the analysis on singular objects.This led me to a discovery of fundamental links between algebraic, analytic and geometric aspects of singularities, particularly in solutions of longstanding problems posed by Whitney, Thom and Hironaka - the originators of the singularity theories in geometry and algebra. This also led me to an extension of classical Sobolev-Nirenberg and Bernstein-Markov inequalities to a singular setting, and to a discovery of `tame' subanalytic sets on which one can do classical local analysis. In the several last years my work resulted in A. a construction of a complete Poincare type Kahler metric off singularities by means of desingularization; B. a discovery of an Euclidean division in dimension larger than one; C. a discovery and a constructive characterization of `universal stratifications'; D. a classification of all `minimal singularities' of Kollar for threefolds. Of course a lot is still left to be done in all these diverse problems. Besides my research I was also fortunate to have three students graduating in the last two years with excellent Ph.D. theses. I surely will continue to work on the problems listed above, however I most of all hope to continue to produce excellent mathematicians using the fertile ground of the diversity of these problems. Finally, in 2011 and 2012 I have proved two well-known longstanding conjectures: 1. posed by Hironaka in 1977 on the Q-universality of resolution of singularities, and 2. Vasconcelos conjecture on a geometric characterization of flatness (fully demystifying this algebraic notion).