The first aspect of our work was to study the question of determination of a metric:
1) on a domain with boundary from the distance between all pairs of boundary points
2) on a closed domain (with no boundary) from the periods of the periodic orbits.
We made several important contribution on these two questions, typically when the metric has « chaotic » properties, that is its distance minimizing curves (called geodesics) are producing a chaotic dynamical system in phase space. The main result we proved in IPFLOW in this direction was with Lefeuvre: we showed that when two metrics with negative curvature are close one to each other and the ordered periods of their periodic orbits are the same, the two metrics are the same up to change of coordinates. This solves partially a famous conjecture from 1985 by Burns and Katok.
Another aspect of our project was the study of resonances and periods of periodic orbits of chaotic dynamical systems, in particular via the use of the zeta function Z(s), a natural function similar to Riemann zeta function but constructed out of periodic orbits of our system. One of the main result in this direction was proved in collaboration with Dang, Rivière and Shen, solving in dimension 3 (and several other cases in higher dimension) an old conjecture by D. Fried saying that the value of the zeta function at s=0 is a purely topological quantity (thus not depending in the dynamical system) called torsion.
For particular spaces enjoying with many local symmetries, called locally symmetric spaces, we also discovered several classical/quantum correspondences, namely a correspondence between the resonances of their geodesic flow and the eigenvalues of the Laplace operator (the eigenfrequencies for the solution of the wave equations). With Bonthonneau, Hilgert and Weich we also developed a completely new theory for defining and studying resonances of an important class of dynamical systems called Anosov actions.
Finally, the spectral and scattering methods used in our IPFLOW project allowed me to solve, with Kupiainen, Rhodes and Vargas, a fundamental problem in conformal field theory in dimension 2, namely the mathematical construction and resolution of the Liouville conformal field theory.Conformal Field Theories are quantum field theories introduced in physics in the early eighties, that appear as scaling limits of certain models in statistical physics. They enjoy a large group of symmetries and have been studied in mathematics using algebraic methods. However, most of these theories are still not solved at the mathematical level. Our result was considered by Quanta Magazine as one of the 3 mathematical breakthroughs in mathematics of the year 2021.
