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 and*its applications are central to my work.*******In the past years my research led me to discovery of fundamental links between algebraic, analytic*and 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 and*Gagliardo-Nirenberg type inequalities of analysis; to sharp new bounds on the heat kernel via*desingularizing weights, to tangential Markov inequalities on singular varieties, to a Chow type*theorem for ideals and by means of the latter and desingularization to a simple construction of a*Poincare type metric off singularities.*******Within the last 6 years I discovered a Bertini-type theorem crucial for my characterization of*Universal Stratifications satisfying Thom and Whitney-a conditions, established complexity bounds*for classical desingularization (turned out to be very high) and low complexity bounds (polynomial)*for Normalized Nash Desingularization in essential dimension 2 , proved Geometric Auslander*criteria 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 desingularizations*isomorphic off these singularities, e.g. just the Whitney Umbrella for surfaces). Recently I also*established 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 and****Malgrange 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 the*information 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-times*differentiable 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 a*fertile ground for raising excellent mathematicians from graduate students. My research plans*aim to repeat this experience.***