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Inverse Problems and Flows

Periodic Reporting for period 4 - IPFLOW (Inverse Problems and Flows)

Reporting period: 2021-07-01 to 2022-12-31

Inverse problems is an area of mathematics and applied mathematics where we know certain measurements on a system and we want to determine the whole system. The system can typically be an equation such as the wave equation in a bounded domain, and we aim at recovering the speed of propagation of waves from the boundary values of all (or many) solutions. Another case of interest to us
is when the system is a metric (i.e. a way to measure the distance between any two points) on a bounded
domain with boundary and we ask if the distance between all pairs of boundary points can determine the metric inside the domain. This last question is related to a famous rigidity problems in geometry: «  can we hear the shape of a drum ? » where we want to determine a domain from its eigenfrequencies. It is also closely linked to the question of determining a metric on a closed finite dimensional space from the lengths of periodic geodesics (the curves minimizing the lengths).

All these problems can be approached by a combination of analysis, geometry and dynamical systems tools. Our first objective is to make significant advances on these questions when the underlying system has certain chaotic properties, by exploiting new analytic methods for studying the long time dynamics of chaotic dynamical systems called Anosov or Axiom A. This long time dynamical behavior can be understood by spectral approaches. The spectrum of the generator of the dynamics is called resonance spectrum and the speed of mixing and convergence to equilibrium is governed by the localization of the resonances.
The resonances are in turn strongly related to the periods of the periodic orbits of the system. An efficient way to study the periodic orbits is to cook up a well-chosen zeta function similar to the famous Riemann zeta function but replacing the prime numbers by the periods of the prime periodic orbits of our system.
The zeros of this zeta function turn out to be given by the resonances, providing a link between periodic orbits and the spectrum of the generator of the dynamics. Our second objective is to study the resonances of a chaotic system and its zeta function. A particular case of interest are the so-called locally symmetric spaces (those admitting many local symmetries) where one has tools coming from harmonic analysis and representation theory.
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.
Our result with Lefeuvre on the determination of the metric from the period of its periodic geodesics on a negatively curved space is an important progress on the understanding of periodic orbits as a way to characterize a space. The introduction of new analytic tools based on microlocal analysis were key for these achievements. These analytic tools also allowed us to make significant progress (with a full resolution in dimension 3) on the Fried conjecture, relating the value at 0 of the zeta function and a topological quantity called torsion. Until our result, the Fried conjecture was only solved for particular cases and for spaces with many local symmetries, known as locally symmetric spaces.
Another important contribution we made with Bonthonneau, Hilgert and Weich was the construction of a new theory of resonances for certain systems called Anosov actions and particularly studied in the area of dynamical systems.
This construction allowed us to construct new invariant measures and should produce new powerful analytic tools to understand these Anosov actions.
Finally, our resolution of the conformal bootstrap conjecture in 2 dimensional conformal field theory, with Kupiainen, Rhodes and Vargas, has pushed the mathematical understanding of non-rational conformal field theory far beyond the state of the art.
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