Program

Integral geometry is the branch of geometrical analysis that studies integral transforms of geometrical nature. The first example is the Radon transform, which transforms functions to their integrals over hyperplanes. Most of the transforms of the modern integral geometry are variations of the Radon transform. The basic idea in integral geometry is the 19th century geometric idea of the incidence between manifolds of geometric objects. The theory of the Radon transform contains remarkable explicit formulas, starting with the inversion formula. The central problem of integral geometry is to find geometrical structures that allow one to develop a similar explicit analysis. Such structures are known on some homogeneous manifolds in the presence of sufficiently many group symmetries. Gelfand and Graev showed, for complex semisimple Lie groups and some other homogeneous spaces, that there is an integral transform of Radon type (the horospherical transform) whose inversion problem is equivalent to the Plancherel Formula. This connection with representation theory is one of principal stimuli for the modern development of integral geometry. But integral geometry also has essential connections with many other areas of mathematics. Some such areas are symplectic geometry, multidimensional complex analysis, algebraic analysis, nonlinear differential equations, and aspects of Riemannian geometry. The Radon transform and its variations are the mathematical base of computer tomography. The program will be concentrate on a few directions where it is possible to expect strong new ideas and results:integral geometry and theory of representationsintegral geometry and multidimensional complex analysisideas of symplectic geometry and algebraic analysis in integral geometrygeometrical structures in integral geometry and non linear differential equations. Many of participants in the proposed program are mathematicians whose research efforts were, and are, important in developing these directions in integral geometry. Sept 3-14, 2001 Special activity on p-adic integral geometry, with a lecture series organized by Alexander Goncharov. Nov 19-30, 2001 Special emphasis on the D-module approach to classical integral geometry, with a series of lectures on D-modules and integral geometry, organized by Alexander Goncharov.