Desingularization and applications. Analysis on and Geometry of singular spaces

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Singularities express irregularities of form in many branches of mathematics and its applications.The important features of forms are often concentrated at singularities. Desingularization andits applications are central to my work.In the past years my research led me to discovery of fundamental links between algebraic, analyticand geometric aspects of singularities: solutions of the long-standing problems posed by Whitney,Thom and Hironaka; to an extension to a singular setting of the classical Bernstein-Markov andGagliardo-Nirenberg type inequalities of analysis; to sharp new bounds on the heat kernel viadesingularizing weights, to tangential Markov inequalities on singular varieties, to a Chow typetheorem for ideals and by means of the latter and desingularization to a simple construction of aPoincare type metric off singularities.Within the last 6 years I discovered a Bertini-type theorem crucial for my characterization ofUniversal Stratifications satisfying Thom and Whitney-a conditions, established complexity boundsfor classical desingularization (turned out to be very high) and low complexity bounds (polynomial)for Normalized Nash Desingularization in essential dimension 2 , proved Geometric Auslandercriteria for openness and for flatness of algebraic morphisms and also advanced my 15 years old`geometric minimal models' program to a classification of `minimal singularities' in 4 and 3 variables(i.e. a minimal list of singularities besides normal crossings with existence of desingularizationsisomorphic off these singularities, e.g. just the Whitney Umbrella for surfaces). Recently I alsoestablished desingularization of the cotangent bundle of singular threefolds (and, in any dimension,its equivalence to a desingularization of the induced metric) and extended my arcanalyticity andMalgrange type division by analytic functions results to the quasi-analytic classes. I plan to work on natural extensions of these results.The main objective of the proposed research is to find a closer link between desingularization and theinformation that may be encoded in the geometry of and analysis on singular spaces. My longer term objective would be a reconstruction of the entire process of resolution of singularities in terms of geometry and/or analysis. I also would like to show that complex analytic stable leaves of a `rigid' Morse-Smale complexes of projective algebraic manifolds extend to algebraic subvarieties of the same dimension, and that function on a closed set which is separately a restriction of a semi-algebraic function and of a k-timesdifferentiable function is a restriction of a k-times differentiable semi-algebraic function, etc. Finally, my attempts to clarify diverse problems in algebra, geometry and analysis proved to be afertile ground for raising excellent mathematicians from graduate students. My research plansaim to repeat this experience.