Micro-local analysis applied to geometry and mathematical physics

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The first objective of this research project is to apply techniques of micro-local analysis in geometry and mathematical physics. The first problem we plan to investigate concerns the asymptotic properties of eigenvectors associated to self-adjoint operators: associated distributions and their relations with the classical flow.A second aspect is the study of statistical properties of the discrete spectrum in relation with the classical dynamics. This question has some important motivations in geometry, physics and in quantum chemistry where the semi-classical approximation and high-energy methods are intensively used. In mathematics several conjectures are open concerning the universal statistics of large eigenvalues for integrable and chaotic motions and we hope to contribute.