Simple Models in Mathematical Physics

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The project is concerned with three simple models in mathematical physics: random matrices and the linear and non-linear Schrodinger equations. On the first topic, the main goal is to demonstrate the appearance of random matrix statistics in models with comparatively little randomness. Investigations of the (linear) Schrodinger equation will center around the forward/inverse spectral/scattering problems for perturbations of periodic potentials. Global well-posedness of the non-linear Schrodinger equation at critical regularity will be attacked, on the torus and in the focusing case.Consider a drum, not necessarily circluar in shape. When struck, the sound produced is built up of infinitely many components at different frequencies; the exact frequencies are determined by the shape of the drum. (The same is true of other instruments; with differing combinations of frequencies responsible for the characteristic sound of each instrument.) From the list of frequencies, one may determine the area and perimeter of the drum, for example, but not the shape (at least in general). On the basis of numerical investigations, two exciting new conjectures have been made about these characteristic frequencies: when the drum has symmetries, they behave very erratically; while in the absence of symmetries the spacing becomes much more regular. Both statements have precise formulations due to Berry and Tabor in the first instance and Bohigas, Giannoni, and Schmit in the second. A large part of this project is devoted toward making progress on the second conjecture. In particular, in the case of a drum chosen 'at random'.