Research
My research interests lie in between geometry, topology and dynamical systems with a focus on problems mixing this three domains, either in the origin of the questions, or in the proofs.
My main two axes of research are concerned with Anosov flows on one side and the geometry and dynamics of Finsler (and Riemannian) metrics on the other.
The unifying theme that underlies my research is a connection between geometry, topology and dynamics: I use geometrical or topological tools to study dynamical systems and study dynamical flows coming from a geometrical setting.
If you'd like to know more, you can have a look at my CV, or at my publications below.
Publications and Preprints
(With Sergio Fenley and Kathryn Mann) Anosov flows with the same periodic orbits (2023) (arxiv)
(With Steven Frankel and Kathryn Mann) Orbit equivalences of pseudo-Anosov flows (2022) (arxiv)
(With Kathryn Mann, appendix with Jonathan Bowden) Orbit equivalences of ℝ-covered Anosov flows and hyperbolic-like actions on the line (2021), accepted in Geometry & Topology (arxiv)
(With Sergio Fenley, and Rafael Potrie) Collapsed Anosov flows and self orbit equivalences (2020), accepted in Commentarii Mathematici Helvetici (arxiv)
(With Andrey Gogolev) Centralizers of partially hyperbolic diffeomorphisms in dimension 3, Discrete Contin. Dyn. Syst. 41 (2021), no. 9, 4477--4484. (arxiv)
(With Sergio Fenley, Steven Frankel, and Rafael Potrie) Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems 41 (2021), no. 11, 3227--3243. (arxiv)
(With Sergio Fenley, Steven Frankel, and Rafael Potrie) Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part II: Branching foliations (2020), accepted in Geometry & Topology (arxiv)
(With Sergio Fenley, Steven Frankel, and Rafael Potrie) Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamical coherent case (2020), accepted in Annales Scientifique de l'ENS (arxiv)
(With Alena Erchenko) Geometry and entropies in a fixed conformal class on surfaces, Annales de l'institut Fourier (Grenoble) 71 (2021), no. 2, 731-755. (arxiv)
(With Sergio Fenley, Steven Frankel, and Rafael Potrie) Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds (Research Announcement), 2018 MATRIX annals, 341–357, MATRIX Book Ser., 3, Springer, Cham, [2020] (arxiv)
(With Christian Bonatti, Andrey Gogolev, and Federico Rodriguez Hertz) Anomalous Anosov flows revisited, Proceedings of the London Mathematical Society Vol. 122, No. 1, 93--117, (2021) (arxiv)
(With Alena Erchenko) Flexibility of geometrical and dynamical data in fixed conformal classes, accepted in Indiana University Mathematics Journal (2017) (arxiv)
(With Andrey Gogolev) A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows, MRL Vol. 26, No. 3, 711-728, 2019 (arxiv)
(With Ludovic Marquis, and Andrew Zimmer) Entropy rigidity of Hilbert and Riemannian metrics, IMRN (2017), no. 22, 6841–6866 (arxiv)
(With Sergio Fenley) Counting orbits of Anosov flows in free homotopy classes, Commentarii Mathematici Helvetici 92 (2017), no. 4, 641–714. (arxiv)
On deformations of the spectrum of a Finsler--Laplacian that preserve the length spectrum, Accepted to the Bulletin de la SMF (2016) (arxiv)
(With Bruno Colbois, Mickaël Crampon and Patrick Verovic) Laplacian and spectral gap in regular Hilbert geometries, Tohoku Mathematical Journal (2) 66 (2014), no. 3, 377-407. (arxiv, journal)
(With Sergio Fenley) Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits, Journal of Topology (2014), no. 3, 677–696. (doi:10.1112/jtopol/jtt041, arxiv)
(With Bruno Colbois) Eigenvalue control for a Finsler-Laplace operator, Annals of Global Analysis and Geometry 44 (2013), no. 1, 43–72.(journal, arxiv)
A Natural Finsler-Laplace operator, Israel Journal of Mathematics, 196 (2013), no. 1 (doi:10.1007/s11856-012-0168-z ,arxiv)
A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows, my dissertation, done under the direction of Patrick Foulon & Boris Hasselblatt, 2012 (pdf, arxiv, TEL)
Not to be published preprint
Anosov flows in 3-manifolds (Preliminary version), lecture notes for the School on contemporary dynamical systems, held in July 2017 at Montréal.
Collaborators
Here's a list of the colleagues I am, or have been, working with:
Christian Bonatti, Université de Bourgogne
Jonathan Bowden, Universität Regensburg
Bruno Colbois, Université de Neuchâtel
Alena Erchenko, Stony Brook University
Sergio Fenley, Florida State University
Patrick Foulon, CIRM
Steven Frankel, Washington University
Andrey Gogolev, Ohio State University
Boris Hasselblatt, Tufts University
Ludovic Marquis, Université de Rennes
Kathryn Mann, Cornell University
Rafael Potrie, Universidad de la Republica Uruguay
Federico Rodriguez Hertz, Pennsylvannia State University
Patrick Verovic, Université de Savoie
Andrew Zimmer, Louisiana State University
Talks
If you want to see a list of talks that I have given, you can look in my CV. But here's some slides that I made for some talks:
Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits, Les Systèmes d'Anosov, CIRM, June 18-22, 2012
Isotopy class for orbits of skewed R-covered Anosov flows, JMM 2012, Special Session on Uniformly and Partially Hyperbolic Dynamical Systems
Un nouvel opérateur de Laplace en géométrie de Finsler et orbites périodiques de flots d'Anosov, my thesis defense