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.
- 28/05-02/06/23: Workshop on Analytic techniques in Dynamics and Geometry (organizer), Les Diablerets, Switzerland.
- 03-07/07/23: IMJ-PRG Summer school on Microlocal and probabilistic methods in Dynamics and Geometry (organizer), Jussieu, France.
- 12/05/23: With Alena Erchenko, we just uploaded a paper on the arXiv in which we prove the marked boundary rigidity of surfaces of Anosov type. This generalizes to surfaces with non-trivial topology the celebrated rigidity result of simple disks by Pestov and Uhlmann. The paper is based on a new transfer principle showing that the marked length spectrum rigidity of closed Anosov manifolds implies the marked boundary rigidity of open manifolds which admit an isometric embedding into a closed Anosov manifold. Such an isometric embedding was established by A. Erchenko with coauthors (D. Chen, A. Gogolev) in certain cases — in particular, for open surfaces of Anosov type — while the marked length spectrum rigidity of closed Anosov surfaces was proved recently by C. Guillarmou, G. Paternain and myself.
- 07/05/23: Je suis co-signataire, avec 188 autres scientifiques, d’une tribune dans le journal Le Monde appelant les actionnaires de TotalEnergies à voter contre la stratégie climat de la firme, et notamment à exiger qu’elle abandonne le projet Eacop de pipeline en Ouganda et en Tanzanie. N’hésitez pas à la relayer !
- 03/05/23: With Mihajlo Cekić, Andrei Moroianu and Uwe Semmelmann, we just uploaded on the arXiv a short paper in which we establish the correspondence between the Weitzenböck identity and the localized Pestov identity. These are two important identities in Riemannian geometry which are used in very different contexts (vanishing of Betti numbers, eigenvalues estimates on the one hand, tensor tomography, geometric/spectral rigidity on the other hand). The aim of the paper was to exhibit the dictionnary relating these two identities; this was probably well-known to some people, but not written anywhere!
- 22/03/23: Two new papers on the arXiv today!
- First, with Colin Guillarmou and Gabriel Paternain, we prove the rigidity of the marked length spectrum for Anosov surfaces namely, we show that two Anosov metrics with same marked length spectrum are isometric (by an isometry isotopic to the identity). This extends to the Anosov setting the celebrated results by Croke and Otal in negative curvature. The proof rests on a new notion of generalised intersection number for an algebra of fiberwise holomorphic distributions invariant by the geodesic flow. The techniques are based on a combination of microlocal analysis and geometry of complex curves (and some mapping class group theory towards the end!).
- Second, with Mihajlo Cekić, we establish stability estimates for the Holonomy Inverse Problem on negatively-curved manifolds. The Holonomy Inverse Problem consists in recovering a connection from the knowledge of the traces of its holonomies along closed geodesics. Injectivity results were contained in two of our former papers (The Holonomy Inverse Problem, to appear in JEMS, and On polynomial structures over spheres) and the present paper contains stability estimates for this problem. The main new ingredient is an approximate non-Abelian Livsic Theorem in hyperbolic dynamics which is of independent interest. This result is a generalization to the non-Abelian setting of the approximate Abelian Livsic Theorem proved by Gouëzel and myself in Classical and microlocal analysis of the X-ray transform on Anosov manifolds.
- 08/03/23: The Peccot lectures on Dynamics and geometry in negative curvature: progress and perspectives start tomorrow at the Collège de France. On that occcasion, I am releasing on my webpage the current version of the manuscript Microlocal analysis in hyperbolic dynamics and geometry that I am currently writing, see here. It will be used as material for the course:
– Lecture 1: Introduction, geometry/analysis on the unit tangent bundle.
– Lecture 2: Hyperbolic dynamics, linear rigidity of the marked length spectrum.
– Lecture 3: Microlocal analysis, spectral theory of Anosov flows.
– Lecture 4: Nonlinear rigidity of the marked length spectrum.
- 19/01/23: With Jan Bohr and Gabriel Paternain, we have just uploaded on the arXiv our new paper on invariant distributions and the transport twistor space of closed surfaces. The purpose of this paper was twofold: first, to introduce the transport twistor space of a closed surface (following the approach of Bohr-Paternain for simple surfaces), which is a complex surface diffeomorphic to the unit ball bundle over the surface equipped with a \dbar operator degenerating to the generator of the geodesic flow on the boundary (the unit tangent bundle); second, to show that fiberwise holomorphic invariant distributions on the unit tangent bundle — which play a fundamental role in tensor tomography and solving the X-ray transform injectivity on the surface — correspond to traces of genuine holomorphic function on twistor space (in the same way as distributions on the real line are boundary values of holomorphic functions on the upper half-plane). This twistor space is a 2—1 branched cover over the (compactification of) the orientation compatible linear complex structures on the tangent bundle.
- 18/01/23: We have just upoaded our new paper with Mihajlo Cekić, Andrei Moroianu and Uwe Semmelmann on the ergodicity of unitary frame flows on Kähler manifolds of negative sectional curvature and even complex dimension. This corresponds to the frame flow along geodesics restricted to frames compatible with the complex structure. In the same spirit of our first paper on frame flow ergodicity, we show that there is a holomorphic pinching (~0.9) such that pinched manifolds have an ergodic unitary frame flow. This extends results by Brin and Gromov.
- 19/12/22: The manuscript of my Habilitation à Diriger des Recherches on Microlocal analysis in hyperbolic dynamics and geometry is available (still a preliminary version, comments are welcome!).
- 08/12/22: I am co-organizing 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 minicourses (by Colin Guillarmou, Malo Jézéquel, and Jared Wunsch) complemented by some research talks. Registration for the summer school just opened here! There will be a poster session for Ph.D students. If you want to present a poster, you can submit it to my email address.
- 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.