program:

* Wednesday 7th, 2pm, IMO salle 3L15: Malo Jézéquel (Paris 6): Local and global trace formulae for smooth hyperbolic diffeomorphisms.

* Thursday 8th, 10am, IMO salle 2L8: Mikko Salo (Univ. Jyvaskyla): Carleman estimates for the geodesic X-ray transform

* Thursday 8th, 2pm, IMO salle 1A8: Frédéric Faure (Univ. Grenoble Alpes): Bounds on the number of Ruelle resonances for Anosov flows.

* Friday 9th, 10am, IMO salle 2L8: Thibault Lefeuvre (Univ. Paris Sud): Boundary rigidity for asymptotically hyperbolic surfaces.

* Friday 9th, 2pm, IMO salle 2L8: Nguyen Viet Dang (Univ. Lyon): Ruelle resonances for Morse-Smale flows.

Abstracts:

M. Salo: Carleman estimates for the geodesic X-ray transform

The standard X-ray transform, where one integrates functions over straight lines, is a well-studied object and forms the basis of medical imaging techniques such as CT and PET. This transform has useful generalizations involving other families of curves, weight factors, and integration of tensor fields. These more general transforms come up in seismic imaging (travel time tomography), in medical imaging (SPECT and ultrasound), and in the mathematical analysis of other inverse problems. Recent work of Guillarmou and others has established a connection between the geodesic X-ray transform and microlocal analysis of flows.

We will discuss the possibility of applying the Carleman estimate methodology from PDE theory to the study of geodesic X-ray transforms. This is joint work with Gabriel Paternain (Cambridge).:

M. Jézéquel: Local and global trace formulae for smooth hyperbolic diffeomorphisms.

F. Faure: Fractal upper bound for the density of Ruelle Pollicott resonances of Anosov flows

Uniformly hyperbolic dynamics (Anosov, Axiom A) have "sensitivity to initial conditions" and manifest "determinist chaotic behavior", e.g. mixing, statistical properties etc. The generator of the evolution operator (the "transfer operator") has a discrete spectrum, called "Ruelle-Pollicott resonances" which describes the effective convergence and fluctuations towards equilibrium. We will present a method of analysis using decomposition into wave packets (or wavelets) that gives new results. This method is similar to the "Weyl-Hörmander phase-space metric method". Joint work with Masato Tsujii.

T. Lefeuvre: Boundary rigidity for asymptotically hyperbolic surfaces.

Given (M,g) a compact manifold with boundary, a natural question that arises in Riemannian geometry is wether one can reconstruct the metric g from the knowledge of the geodesic distance between any pair of boundary points. In 1981, R. Michel conjectured that a simple metric is boundary rigid, i.e. that it is determined by its boundary distance up to a natural obstruction, namely diffeomorphisms leaving the boundary invariant. We will survey some of the results of rigidity and present a new one, relative to the marked boundary rigidity of asymptotically hyperbolic surfaces.

N.V. Dang: Ruelle resonances for Morse-Smale flows.