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Abstract: The course is an overview of techniques developed in the last two decades for the mathematical analysis of physical processes taking place in "thin" bodies (such as elastic rods, plates and shells, or quantum wires). Physics experiments have demonstrated a range of unusual phenomena in thin structures, which can be understood by using various mathematical approaches: multi-scale asymptotic analysis, the theory of representations for linear operators in Hilbert spaces ("functional model"), calculus of variations. The course is meant to be an introduction to the field and the related literature, which should inform the participant of the state of the art.
Abstract: The last few decades has seen an explosion of activity in the Physics community based on the premise that one can construct composite media with unusual wave properties: for example the fabrication of composite lens capable of resolving objects below the diffraction limit. Such effects can be corroborated and explained by studying the coupling of different scales present in a given system by non-standard homogenisation methods. The challenges presented in such a mathematical study requires one to revisit, in the multi-scale context, fundamental concepts such as compactness and convergence statements in standard operator, spectral and variational theories. This course focuses on some recent mathematical acitivies in the two-scale asymptotic methods to study multi-scale couplings in `partially degenerate' elliptic PDEs.
Abstract: This course deals with the mathematics behind an infinite mass-spring linear system which is used to model linear crystals. On the basis of Newton's dynamical equations, assuming that the system is within the regime of validity of the Hooke law, one constructs a self-adjoint operator. The movement of the system is a superposition of harmonic oscillations whose frequencies are given by the spectrum of the self-adjoint operator. In this course, we consider families of finite-rank perturbations of the operator corresponding to changes in spring constants and masses of the system. We study how the spectrum changes under such perturbations and consider the inverse problem of finding the masses and spring constants of the system from the spectrum of two elements of the family of perturbed operators.
Abstract: Scattering theory plays a prominent role in virtually all areas of pure and applied physics. A great deal of the knowledge that we have about physical systems was obtained by means of scattering experiments. In particular, inverse scattering theory has many applications in tomography, non-destructive testing, geophysical prospection, etc.
I will give an introduction to this fascinating subject. I will discuss, in a general setup, the main objects of (direct) scattering theory, the unperturbed and the perturbed Hamiltonians, the wave operators, the scattering operator and the scattering matrix. I will consider their main properties, the intertwining relations, the completeness of the wave operators, the unitarity of the scattering operator, and the differential cross section, that is actually measured in the scattering experiments. I will illustrate these results in the particular case of potential scattering in quantum mechanics, where I will discuss the inverse problem of uniquely reconstructing the potential from the high-velocity limit of the scattering operator.
The topics in mathematical scattering theory that I will present are based in perturbation theory of operators in Hilbert space. However, the necessary background material will be given in the lectures.
