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
April 15. (IHP, Salle Yvette Cauchois) — 14h
14h. Carlos Matheus (Centre de Mathématiques Laurent Schwartz)
Mixing rates for geodesic flows on non-positively curved surfaces
In this talk, we shall discuss statistical laws satisfied by the geodesic flows on certain classes of non-positively curved surfaces. In particular, we will see how these laws are affected by the shape of the zero curvature regions. This is based on joint works with Y. Lima and I. Melbourne.
15h30. Daniel Monclair (Université Paris-Saclay)
Spectra of anti-de Sitter quasi-Fuchsian manifolds
Two spectral theories arise from the study of hyperbolic manifolds: the Laplacian, and the geodesic flow (Ruelle-Pollicott resonances). There are many results building explicit bridges between these two spectra. For Lorentzian manifolds, the elliptic Laplacian is replaced with a hyperbolic operator, leading to a completely different spectral theory. We will see that we can still connect this operator with the geodesic flow for some 3-dimensional anti-de Sitter (i.e. Lorentzian with constant negative curvature) manifolds.
Based on joint work with B. Delarue and C. Guillarmou.
May 13. (IHP, Salle Yvette Cauchois) — 14h
Past sessions
. March 18 (IHP, Salle Yvette Cauchois) — 14h
14h. Marco Mazzucchelli (Sorbonne Université)
Length Spectrum Rigidity and Flexibility of Spheres of Revolution
It is well known, since the classical work of Darboux and Zoll, that the length spectrum of a Riemannian 2-sphere does not provide sufficient information to recover the Riemannian metric. In this talk, based on joint work with Alberto Abbondandolo, I will introduce a suitable notion of marked length spectrum on each S^1-symmetric Riemannian metric on the 2-sphere having only one equator, and study the rigidity and flexibility imposed by these data.
While the marked length spectrum does not recover the metric, I will show that it recovers at least the contact geometry of the unit tangent bundle: isospectral metrics have conjugate geodesic flows.
Under a further Z_2-symmetry assumption, the marked length spectrum does recover the metric. Indeed, every isospectral class of metrics contains a unique Z_2-symmetric metric, and I will give an explicit description of every isospectral class as an infinite-dimensional convex set, generalizing the known description of S^1-symmetric Zoll metrics.
15h30. Karen Butt (University of Chicago)
Marked Poincare rigidity
Given a closed negatively curved manifold, we consider the extent to which dynamical data associated to its closed geodesics (equivalently, periodic orbits of its geodesic flow) determines the underlying metric up to isometry. For instance, the lengths of closed geodesics, marked by their free homotopy classes, are conjectured to characterize the underlying metric up to isometry. In this talk, we consider a dynamically flavored variant of this marked length spectrum rigidity problem. We introduce the marked Poincare determinant, which associates to each free homotopy class of closed curves a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. Our main result is that near hyperbolic metrics in dimension 3, this invariant determines the metric up to homothety. This is joint work with Erchenko, Humbert, Lefeuvre, and Wilkinson.
. February 11 (IHP, Amphithéâtre Choquet-Bruhat) — 14h
14h. Jared Wunsch (Northwestern University)
Scattering of internal waves in a two-dimensional channel
I will describe joint work with Zhenhao Li and Jian Wang on the scattering of internal waves in a two-dimensional channel with flat ends. We obtain the uniqueness of appropriately defined outgoing solutions, and consequently show that a scattering matrix exists and may be regarded as a smoothing perturbation of a « bounce map » obtained from ray-tracing. We moreover obtain a limiting absorption principle which allows us to describe long-time asymptotics of of solutions.
15h30. Jayadev Athreya (University of Washington)
Semiclassical measures for complex hyperbolic quotients
In joint work with Semyon Dyatlov and Nicholas Miller, we study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-Jézéquel on quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.
. January 21 (IHP, Amphithéâtre Choquet-Bruhat) — 14h
14h. Davide Tramontana (University of Bologna)
Subelliptic random walks on Riemannian manifolds and their convergence to equilibrium
In this talk, we construct a random walk on a closed Riemannian manifold associated with a second-order subelliptic differential operator and prove its convergence to equilibrium. The construction relies on a local reduction to an operator with constant coefficients, using a technique of Fefferman and Phong based on Calderón–Zygmund localization. Convergence to equilibrium is then obtained through the spectral theory of the associated Markov operator.
15h30. Yannick Guedes Bonthonneau (ENS)
Spectrum for Anosov representations
Anosov representations are certain representations of discrete groups in large Lie groups. I will report on work in progress with Lefeuvre and Weich, where we associate these objects with spectral data consisting of complex hypersurfaces in C^k. We rely on a recent innovation enabling the use of tools from smooth hyperbolic dynamics.
. 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.
THIBAULT LEFEUVRE 