The work described in the proposal concerns several topics in Classical Analysis: approximation theory, several complex variables, miscellaneous problems on complex polynomials and classical operator theory. The most central questions for the project are: uniqueness of best approximation by harmonic functions in the supremum norm in dimensions three and higher (the two-dimensional case has been recently settled by the PI and H.S.Shapiro), sharp estimates relating to multidimensional analogs of H. Bohr's phenomenon for power series, counting zeros of complex-valued harmonic polynomials, Agler-Cole-Wermer interpretation of Ando's theorem in C^2 and, finally, understanding the spectral picture of the celebrated double layer potential operator -a topic going back to the pioneering work of I. Fredholm.
Most of the questions in the project are rather simply stated and, accordingly, will attract a fair number of graduate students and young researchers. The problems are also open-ended and suggest several new paths of investigation in these classical areas away from the well traveled tracks. The investigations undertaken in the project have well-defined applications to Mathematical Physics (Potential Theory), Applied Mathematics (polynomials, special functions) and Numerical Analysis (explicit solutions of some of the most important boundary value problems). The project entails active involvement of undergraduate and graduate students at all stages of the project and, via local network of workshops and seminars for teachers, to disseminate some of the more elementary off-beats of the proposed work among high school students interested in science.
