# Thibault Lefeuvre

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.

#### 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.
• 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.