THIBAULT LEFEUVRE

The ADG seminar—Analysis, Dynamics, Geometry seminar— aims to bring together researchers based in and around Paris who are interested in microlocal and semiclassical analysis, and their various manifestations in dynamical systems and geometry (quantum ergodicity, Anosov flows, inverse problems, geometric rigidity, etc.)

The seminar takes place once a month at IHP and consists of two talks.

Organizers: Mihajlo Cekić (Université Paris-Est Créteil) and Thibault Lefeuvre (Université Paris-Saclay). Funded by the ERC Starting Grant ADG (Analytic techniques in Dynamical systems and Geometry).

Upcoming sessions

. January 21 (IHP, Amphithéâtre Choquet-Bruhat) — 14h

Davide Tramontana (University of Bologna)

Yannick Guedes Bonthonneau (ENS)

. February 11 (IHP, Amphithéâtre Choquet-Bruhat) — 14h

Jared Wunsch (Northwestern University)

Jayadev Athreya (University of Washington)

. March 18 (IHP, Salle Yvette Cauchois) — 14h

Marco Mazzucchelli (Sorbonne Université)

Karen Butt (University of Chicago)

Past sessions

. December 17 (IHP, Amphithéâtre Choquet-Bruhat) — 14h

14h. Nicolas Burq (Université Paris-Saclay)

Observation and control for Schrödinger equations

The motivations for this works come from understanding how solutions to some PDE’s or eigenfunctions of Laplace operators can concentrate on small sets. I will describe two kinds of results. First on rational tori when we observe the $L^{^2}$ norms on space time measurable sets and second in more general geometries, under geometric control conditions or on negatively curved compact surfaces when we observe solutions to Schödinger equations on time measurable sets. All the results presented rely on obtaining precise high frequency description/estimates. This is based on joint works with Hui Zhu (NYU Abu Dhabi).

15h30. Matthieu Léautaud (Université Paris-Saclay)

Poincaré series of convex bodies

We consider the set of distances from a point to a lattice in Euclidean space, for a metric related to a convex body. Associated with these lengths, we construct a Poincaré series: a natural holomorphic function defined in a complex half-plane. The aim of the talk is to study this function: its possible extension to the other half-plane, its poles, its singularities, etc. In doing so, we encounter a multiplication operator by a Morse function on the sphere and describe its spectral theory. This is joint work with Nguyen Viet Dang, Yannick Guedes-Bonthonneau, and Gabriel Rivière.

. November 26 (IHP, Amphithéâtre Darboux) — 14h

14h. Giovanni Forni (Cergy Paris Université)

Prequantizing rational billiard flows

We will present some basic results on the prequantized dynamics of rational billiards and translation flows, including minimality, unique ergodicity, (relative) decay of correlations and absolute continuity of the L^2 spectrum. This is joint work with F. Arana-Herrera and J. Athreya.

15h30. Daniele Galli (University of Zurich)

Effective unique ergodicity for translation flows via Microlocal Analysis

The aim of this talk is to describe a new method, based on renormalization and Microlocal Analysis, to recover some celebrated results in the field of Teichmüller Theory of translation surfaces. In particular, we will show how these techniques give a new proof of effective unique ergodicity criterion for the linear flow on translation surfaces, firstly proved by Forni (2002). This is a joint project with Hamid Al-Saqban.

When the boundary is Anosov with simple length spectrum, the study of singularities in the trace of the wave operator reveals certain interesting spectral invariants via the Duistermaat-Guillemin trace formula. We will discuss how these invariants can be exploited and naturally combined with the injectivity of the geodesic X-ray transform to address the problem.

In this context, some recent positive results obtained in the class of conformal metrics will be presented and it will be explained how it is possible to hear the jet at the boundary of a conformal Steklov drum. Finally, we will briefly discuss some results obtained by similar techniques on the magnetic Steklov inverse problem, concerning the recovery of an electric potential and a magnetic field from the spectrum of the associated magnetic DN map.

. November 5 (IHP, Amphithéâtre Darboux) — 14h

14h — Sebastián Muñoz-Thon (Université Paris-Saclay)

Guillarmou’s Pi operator for magnetic and thermostat flows

In the last decade, the study of inverse problems on closed manifolds has become a rising topic. One of the reasons is the breakthrough by Guillarmou and Lefeuvre on the Burns-Katok conjecture: on closed manifolds, for close enough metrics of negative curvature, the marked length spectrum determines the metric up to diffeomorphism isotopic to the identity. In the proof one uses the Pi operator introduced by Guillarmou: it can be defined as (the sum of) the holomorphic part(s) of the Laurent expansion of the resolvent of the generator of the flow, acting on anisotropic Sobolev spaces. The relevance of Pi comes from the fact that it is related to the X-ray transform, i.e., with the linearization of the length. In this talk, I will discuss how to generalize this object to more general flows such as the magnetic and thermostat flows. In particular, we show that Pi is a pseudodifferential operator of order -1, and we will use it to obtain a stability estimate for the linearized problem. This is a joint work with Sean Richardson. 

15h30 — Benjamin Florentin (Université de Lorraine)

Can one hear the shape of a Steklov drum ? The approach of wave trace invariants

Introduced at the beginning of the 20th century, the Steklov eigenvalue problem has attracted growing interest in spectral geometry over the last few decades and remains a major research topic in the field. In this talk, we will focus on the spectral inverse problem consisting in recovering a metric of a compact Riemannian manifold with boundary from knowledge of its Steklov spectrum, or equivalently the spectrum of its Dirichlet-to-Neumann map (DN map). In other words, can one hear the shape of a  » Steklov drum  » ?

October 1 (IHP, Amphithéâtre Darboux) — 14h

14h. Alix Deleporte (Université Paris-Saclay)

Spectral estimates, heat observability, and thickness on Riemannian manifolds

Eigenfunctions of the Laplacian cannot vanish on a set of positive measure. Quantitative versions of this unique continuation are well-known on fixed Riemannian manifolds : the L2 norm of an eigenfunction is bounded by its L2 norm on a set of positive measure times a constant which grows exponentially with the frequency. This growing rate is sharp and reflects in observability properties for the heat equation.

In this talk, I will present recent results, in collaboration with M. Rouveyrol (Uni. Bielefeld) about these questions in a non-compact setting, and/or uniformly with respect to the metric. Quantitative unique continuation, and observability of the heat equation, hold under a necessary and sufficient condition of thickness of the observed set : it must intersect every large enough metric ball with a mass bounded from below, proportionally to the mass of the ball itself. 

I will talk about the case of non-compact hyperbolic surfaces, then about much more general Riemannian manifolds (in progress!). The proof crucially uses the Logunov-Mallinikova estimates.

15h30. Zhongkai Tao (IHES)

Lossless Strichartz and spectral projection estimates on manifolds with trapping

The Strichartz estimate is an important estimate in proving the well-posedness of dispersive PDEs. People believed that the lossless Strichartz estimate could not hold on manifolds with trapping (for example, the local smoothing estimate always comes with a logarithmic loss in the presence of trapping). Surprisingly, in 2009, Burq, Guillarmou, and Hassell proved a lossless Strichartz estimate for manifolds with trapping under the « pressure condition ». I will talk about their result as well as our recent work with Xiaoqi Huang, Christopher Sogge, and Zhexing Zhang, which goes beyond the pressure condition using the fractal uncertainty principle and a logarithmic short-time Strichartz estimate. If time permits, I will also talk about the lossless spectral projection estimate in the same setting.