géométrie et théorie spectrale

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Geometric spectral theory is an area of mathematics that lies at the intersection of analysis, partial differential equations and differential geometry. To a large extent, it is motivated by questions originating in the study of real life phenomena, such as vibration, heat propagation, oscillations of fluids and quantum mechanical effects. The first part of the proposal is concerned with the spectral geometry of the Steklov problem. This eigenvalue problem, introduced by V. Steklov in 1902, has many important applications, but is still relatively unexplored by pure mathematicians. The proposed topics of research include the study of singular Steklov type problems, such as the sloshing problem; isospectrality and spectral invariants; nodal geometry of Steklov eigenfunctions. The second part of the proposal focuses on some questions arising in the study of spectral asymptotics of Laplace and Schrodinger operators.