Mathematical Sciences: Complex Integral Geometry and Analysis at Flag Domains

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Absract Gindikin At the focus of the project there are 2 aspects of integral geometry. First, it is the analysis in flag domains which extends the analysis in noncompact Hermitian symmetric spaces. The first step is the definition of analogs of elementary functions which we call the determinant functions. The well known examples of such functions are the norm-functions for symmetric domains (or Jordan algebras), but for flag domains there is a richer collection of new functions. Using these functions we hope to obtain several explicit formulas and results: descriptions of Stein neighborhood of Riemann symmetric spaces where it might be possible to holomorphically extend solutions of the Schmid equations, parametrizations of complex cycles in flag domains, generalizations of the Hua - Poisson integrals and the Hua equations for them, their computations on the language of integral geometry and multidimensional residues, the generalized Penrose transform etc. The essential role in these constructions are played by the boundary values of the cohomology in nonconvex tube domains. Another direction in this project is an axiomatic of the method of horospheres and its applications. Several years ago in the process of solving the Gelfand problem I gave some axiomatic conditions on a family of submanifolds of a complex manifold providing an explicit local inversion formula of the corresponding problem of integral geometry. These conditions are satisfied for the horospheres on complex semisimple Lie groups, and it is the way to invert the horospherical transform without using group structures. Now we extend this axiomatic in such a way that it becomes possible to invert the horospherical transform for some nonsymmetric homogeneous manifold. The integral geometry is a direction of geometric analysis which connects analysis on manifolds with geometrical structures on them. The philosophy of integral geometry is that there are geometrical structures more general than group invariance which give a base for the development of a rich multidimensional analysis with important applications to analysis on homogeneous manifolds, complex analysis, nonlinear differential equations , mathematical physics, etc. Integral geometry is a theoretical base of computer tomography.