| Integral geometry is a part of mathematics connecting various branches of mathematical analysis, operator theory, algebra, geometry. It has numerous practical applications, for example, to computerized tomography (CAT-scans), inverse problems in geophysics and material science, and to indirect measurement problems. Integral geometry studies certain integral transformations, which assign to a mathematical object defined on a manifold the collection of its integrals along a family of submanifolds. The problem then is to reconstruct the object completely or partially using these integrals as data. The main emphasis in previous developments of integral geometry was on the problems concerning scalar-valued functions. There was also some work on the integral geometry of differential forms and cohomologies. Prof. V. A. Sharafutdinov was the first to systematically study the corresponding questions for symmetric tensor fields in real space. His work originated in the so called photoelasticity problem—the problem of stress tensor reconstruction in a transparent sample using polarization shift data for the rays along some set of directions.; In the present dissertation the author constructs an analogous theory in complex space (chapters 1,2), taking into account the recent appearances of the complex integral transforms in twistor theory, mathematical physics, and group representation theory.; Chapter 2 of the dissertation is devoted to the incomplete data problems for symmetric tensor fields both in real and complex cases.; In chapter 3 a weighted integral geometry problem for matrices, and a nonlinear problem of matrix reconstruction are considered. They represent nontrivial matrix generalizations of the classical Radon transform.; In chapter 4 an inverse problem for a Vlasov system of equations describing stationary plasma is investigated with the help of vector tomography techniques.; In chapter 5 nonuniqueness examples are given for the attenuated (scalar) Radon transforms and Radon transforms with weight invariant under rotations.; In chapter 6, which is the result of a joint work with prof. C. A. Berenstein and prof. L. A. Aizenberg, problems of characterization for pluriharmonic and separately harmonic functions in terms of mean-value theorems are considered.; Several problems of the dissertation remain open. The author believes that this area of research will develop actively and have many new applications. |
