I am a French mathematician, CNRS Junior researcher working at Sorbonne Université (IMJ-PRG), and former student of the Ecole polytechnique. I am also a writer, published by Gallimard. My first novel *Éducation tropicale *was published in 2018 and was awarded the Albert Bernard prize.

#### Upcoming talks:

**10/01/22:**Mon cours*Analyse géométrique sur les variétés / Geometric analysis on manifolds*commencera à Jussieu.**18/01/22:**Séminaire de systèmes dynamiques, Bochum, Allemagne.

#### Latest news:

**11/12/21:**I uploaded on arXiv a note on**isometric extensions of Anosov flows**. This approach is the one that we use in the frame flow paper below. I show that, for any such isometric extension, one can associate a natural representation into the isometries of the fibers, which I call**Parry’s representation**. The idea is then that there exists a**dictionary**between**algebraic properties**of this representation and**dynamical properties**of the extended flow. For instance, the extended flow is ergodic if and only if the representation acts transitively on the fiber. More generally, if the representation preserves a certain structure on the fiber, then this structure will give rise in turn to a flow-invariant structure on the whole bundle.

**29/11/21:**With Mihajlo Cekić, Andrei Moroianu and Uwe Semmelmann, we just uploaded on arXiv a new paper on the**ergodicity of the frame flow on negatively-curved manifolds of even dimension**(and dimension 7). In dimension 4 and 4k+2 (k>1), we almost solve a**long-standing conjecture of Brin**(’70-’80) asserting that the frame flow should be ergodic on 1/4-pinched manifolds: we show that manifolds with**~0.27-pinched curvature**have an ergodic frame flow. In other even dimensions (and dimension 7), the pinching condition that we get is slightly worse and ~0.55. In all cases, this improves by far all the results available in the literature (Brin-Gromov ’80, Brin-Karcher ’83, Burns-Pollicott ’03). The proof combines three main technologies: 1) hyperbolic dynamical systems and the non Abelian Livsic theory developed with M. Cekić in The Holonomy Inverse Problem, 2) the topology of G-structures over spheres, 3) harmonic analysis on the sphere bundle (the twisted Pestov identity).

**13/07/21:**We just uploaded on arXiv our new paper with Mihajlo Cekić on the**generic injectivity of the X-ray transform**. We show that the X-ray transform (namely, the operator of integration of symmetric tensors along closed geodesics) is generically injective on closed Anosov manifolds (i.e. manifolds with hyperbolic geodesic flow such as negatively-curved manifolds) and on certain manifolds with boundary and hyperbolic trapped set. Following earlier results by Guillarmou, Knieper and myself, this solves locally the**marked length spectrum rigidity conjecture**in a neighborhood of a generic Anosov metric. The proof is based on a**perturbation theory**of the zero eigenvalue of elliptic pseudodifferential operators that we develop: the basic strategy is to turn the problem of generic injectivity into an algebraic problem of**representation theory**by using**Gaussian states**.

**12/07/21:**I will be teaching next year a course on*Geometric analysis on manifolds*in the Master de Mathématiques Fondamentales de Jussieu. The lecture notes are under construction but some sections are already available (comments are welcome!): see there. They might eventually be added as an appendix of the manuscript on*Geometric inverse problems on Anosov manifolds*that I am writing.

**14/05/21:**We uploaded yesterday on arXiv our new paper with Mihajlo Cekić on the**holonomy inverse problem**. Given a negatively-curved Riemannian manifold, we consider the restriction of the**Wilson loop operator of field theory**to primitive closed geodesics. This operator, which we call the**primitive trace map**, consists in taking the trace of the holonomy of a unitary connection (on a certain vector bundle) along primitive closed geodesics. We conjecture that in odd dimensions > 2, the primitive trace map should be globally injective on the moduli space of all connections (up to gauge). We prove this result in a lot of situations: for almost-flat connections, for sums of connections on line bundles, and locally near a generic connection. The argument is based on two new ingredients: a Livsic-type theorem in hyperbolic dynamical systems, based on**representation theory**, showing that the cohomology class of a unitary cocycle is determined by its traces along closed geodesics, and a theorem relating the moduli space of connections and the**Pollicott-Ruelle resonances**near zero of a certain natural transport operator.

**14/05/21:**We just uploaded on the arXiv a new version of our paper with Yannick Guedes Bonthonneau on**radial source estimates in Hölder-Zygmund spaces**for hyperbolic dynamics. The main consequence is that we can prove locally the**marked length spectrum rigidity conjecture**of Burns-Katok for metrics which are C^{3+}-close. As a byproduct, we also show that most of the known regularity results in hyperbolic dynamics (in particular those concerning Livsic theory) fit into this general framework of source-type estimates.