Thibault Lefeuvre

© Patrick Imbert, Collège de France

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 events:
latest news:
  • 28/11/22: With Yann Chaubet, Yannick Guedes Bonthonneau and Leo Tzou, we just uploaded our new paper on the arXiv, in which we study geodesic Lévy flights and the narrow capture problem. Lévy flights are stochastic processes on Riemannian manifolds which, unlike Brownian motion, may jump from one point to another along geodesics. They are used in the field of biology to model predators hunting preys: this is known as the Lévy flight foraging hypothesis. We compute the asymptotics of the expected stopping time to find a small target the size of a geodesic ball of radius epsilon (as epsilon goes to zero) in a closed manifold. The proof relies on a precise analytic understanding of the generator of the pure jump Lévy process: we prove that, when the manifold is the sphere, the torus or has negative sectional curvature, it is an elliptic pseudodifferential operator.
  • 27/11/22: I will be giving the 2022-2023 Peccot Lectures in March 2023 on the topic: Dynamics and geometry in negative curvature: new progress and perspectives. I updated the webpage with the content of the course, see here (or tab « Enseignements »).
  • 07/11/22: I am very happy to announce that I was awarded the Brin Prize for Young Mathematicians for my work on dynamical systems at the 33rd Fall meeting of the workshop in dynamical systems at Penn State University! 🙂
  • 22/09/22: We finished writing our new paper with Mihajlo Cekić on polynomial structures over spheres. The paper is dedicated to the memory of Steve Zelditch who passed away on September 11th and was a leading figure in the field of spectral geometry to which this paper belongs. The aim of this article is to explain a relation between three a priori unrelated questions belonging to different fields:
    — In algebraic geometry: the classification of non-trivial polynomial maps between spheres,
    — In spectral theory: the study of isospectral connections (i.e. connections with same spectrum for their Bochner Laplacian), similarly to Kac’s original question for metrics Can one hear the shape of a drum?
    — In dynamical systems: the study of the ergodicity of certain partially hyperbolic flows obtained as isometric extensions over the geodesic flow in negative curvature.
    In particular, we show that, under a low-rank assumption, the spectrum of the Bochner Laplacian fully determines the connection and the topology of the underlying vector bundle.
  • 12/09/22: With Artur Avila and Mihajlo Cekić, I am co-organizing in May 2023 a workshop on Analytic techniques in dynamics and geometry in Les Diablerets (Swiss Alps). The website for the conference is here.
  • 12/09/22: I am very happy to co-organize with Mihajlo Cekić, Oana Ivanovici and Frédéric Naud the 2023 edition of the IMJ-PRG Summer School in Paris (July 2023). This year, the topic will be Microlocal and probabilistic methods in dynamics and geometry. There will be four minicourses (by Colin Guillarmou, Malo Jézéquel, Laura Monk and Jared Wunsch) complemented by some research talks. Registration for the summer school just opened here!
  • 13/05/22: We completely rewrote our paper with Yannick Guedes Bonthonneau on the marked length spectrum conjecture for manifolds with cusps, entitled Local rigidity of manifolds with hyperbolics cusps II. Nonlinear theory and just uploaded the new version to arXiv. There are major changes: first of all, we prove the expected theorem in full generality, that is, we show that the marked length spectrum locally determines the metric for cusp manifolds. Secondly, we clarify the exposition, especially the sections relative to the microlocal calculus tailored for cusp manifolds that we developed in the first paper Local rigidity of manifolds with hyperbolics cusps II. Linear theory and microlocal tools. Last but not least, the proof also relies on the meromorphic extension of the resolvent of the geodesic flow in Hölder-Zygmund regularity. Even for compact manifolds, and to the best of our knowledge, this was previously unknown. This should have interesting consequences, like some good regularity properties for SRB measures for instance.

A surface with three cusps. In red, a closed geodesic in a hyperbolic free homotopy class. In blue, a curve in a free homotopy class of loops wrapping once around a cusp: this class does not contain any closed geodesic.
  • 12/04/22: We just finished writing a short survey paper with Mihajlo Cekić, Andrei Moroianu and Uwe Semmelmann on frame flow ergodicity and, more generally, on ergodicity for certain isometric partially hyperbolic dynamics. Our aim was to explain the circle of ideas in our former paper unlocking Brin’s long-standing conjecture on the ergodicity of the frame flow on 1/4-pinched negatively-curved Riemannian manifolds. It is intended not to be too technical and also gives a broad overview of the historical context.
  • 07/04/22: We just uploaded our new paper with Mihajlo Cekić and Colin Guillarmou on local lens rigidity: we show that, on negatively-curved Riemannian manifolds with strictly convex boundary, the lens data, namely, the lengths of geodesics joining pairs of boundary points together with their incoming/outgoing vectors, locally determine the metric up to isometries. The proof is technical and based on Pollicott-Ruelle theory together with complex interpolation. The main obstacle to overcome is the lack of marking which makes this problem similar to the unmarked length spectrum rigidity on closed manifolds. This provides the first general result on lens rigidity with trapping in dimension greater than 2.