Integral geometry deals with those integral transforms which associate to “functions” on a manifold X their integrals along submanifolds parameterized by another manifold Ξ. 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.
Sheaves and D-modules in integral geometry
D'AGNOLO, ANDREA
2000
Abstract
Integral geometry deals with those integral transforms which associate to “functions” on a manifold X their integrals along submanifolds parameterized by another manifold Ξ. 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.
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