Combinatorical and Algebraic Structures Seminar

Session details

Date: 3.3.2006
Speaker: Jean-Pierre Gazeau, APC, Université Paris VII
Title: Mathematics of the Diffraction
Abstract: It is well known that the diffraction pattern of a periodic structure is periodic. This property is based on the ubiquitous Poisson summation formula. Beyond this basic feature of harmonic analysis, the diffraction is a certain fashion to look at a structure made up of scattering centers, like the atoms in a solid material, and the outcome of the diffraction experiment gives an approximate representation of the spatial organization of the real structure. As a matter of fact, the diffraction pattern of a glass is diffuse, thus reflecting the randomness of the atomic position. On the other hand, the diffraction pattern of an aperiodic structure like a Penrose tiling or a quasicrystal shows features which are the mark of a long-range order in the disposition of the scattering centers.
A certain number of rigorous results have been established during the last ten years in mathematics of the diffraction. Proofs are based on measure and distribution theory and on harmonic analysis.
We propose in this course to give a detailed list of these mathematical results, most of them with proofs, after presenting the mathematical background issued from measure theory (especially translation bounded Radon measures), from distribution theory, and from harmonic analysis combined with the theory of almost-periodic functions. The lessons will be illustrated by various numerical and experimental examples.

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