Micro-local analysis applied to geometry and mathematical physics
Final Report Abstract
We have obtained a more precise description of the duality between quantum spectrum and length spectrum. Via trace formulae, or semi-classical trace formula, we can now obtain statistical results on long periodic orbits. These results are mathematically rigorous and new. Applications of these results in future research should be: - Inverse spectral problems, e.g., what kind of information can we retrieve on the potential from the spectrum of the associated Schrödinger operator? - Application to quantum chaos in various form: study of quantum eigenvectors and their weak limits (as L2-distributions). - Rigorous study of eigenvalues statistics: spectral variance, spectral covailance and spectral correlation. On the other side, I think I have also been partially successful to use methods of semi-classical and microlocal analysis in geometry. We have in particular great hope to obtain nice global results in complex geometry with G. Marinescu. Unfortunately, concerning the question of quanttmr unique ergodicity (in geometry), I can at this moment claim no improvement. Finally, I hope that I have been able to communicate some ideas and techniques from micro-local analysis to mathematicians and young Ph.D. students non familiar with this field. I will continue to communicate in this field with some lectures on trace formulae and associated methods.
