Mathematical Sciences: Singular Integral Operators and Applications

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This project will involve the study of singularities in the theory of singular integrals and differentiation of integrals, on the d-bar equation on domains of finite type in several complex variables, and on dispersive nonlinear partial differential equations, with emphasis on applications of techniques from harmonic analysis and singular integrals. The five major topics to be studied are (1) the d-bar equation on domains of finite type in several complex variables, (2) singular maximal functions and differentiation of integrals, (3) nonlinear dispersive partial differential equations, (4) singular integrals on sets and analytic capacity, and (5) a question in non-commutative ergodic theory. The research in this project is aimed at the study of partial differential equations and the tools used in their analysis. In particular, it involves the applications of the ideas of harmonic analysis to problems in several complex variables and nonlinear dispersive equations.