Sheaves and D-modules in integral geometry

Abstract

Integral geometry deals with those integral transforms which associate to ``functions'' on a manifold their integrals along submanifolds parameterized by another manifold. Basic problems in this context are range characterization--where systems of PDE appear--and inversion formulae. As we pointed out in a series of joint papers with Pierre Schapira, the language of sheaves and D-modules provides both a natural framework and powerful tools for the study of such problems. In particular, it provides a general adjunction formula which is a sort of archetypical theorem in integral geometry. Focusing on range characterization, we illustrate this approach with a discussion of the Radon transform, in some of its manifold manifestations.