Oskolkov will continue his investigations of oscillatory sums and integrals involving imaginary exponentials of real algebraic polynomials. The coefficients of the polynomials are considered as independent real variables. These sums and integrals establish interconnections between such apparently distant fields as Harmonic Analysis, Analytic Number Theory and Partial Differential Equations. He will also elaboratee on Radon-Fourier analysis for functions of several variables. Special emphasis will be made on associated problems of ridge approximation, which consist in selection of best fit linear combinations of planar wave type functions. These problems are of highly non-linear nature, especially with respect to optimal selection of wave vectors. Immediate ramifications of such research include neural network models that have found wide applications in industry, engineering, computer science, finance and medicine. The Radon transform is an acknowledged powerful tool in several applied areas ranging from classical quantum mechanics and optics to X-ray tomography. In computational X-ray tomography, ridge approximation can be interpreted as reconstruction of images of non-homogeneities such as tumors. Another extremely different field is recognition of main trends in evolution of large systems such as bacteria populations or the stock market, where an enormous number of variables must be modeled.
