We discuss inverse problems for nonlinear equations with the common feature that the nonlinear interaction of waves produces new waves that help some inverse problems that remain unsolved for their linear counterpart.

The conjecture broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. The aim of the talk will be to present some very recent results in this direction, as well as extensions to higher dimensions. This is joint work with Mihaela Ifrim.

High energy spectral asymptotics for Schrödinger operators on compact manifolds have been well studied since the early 1900s and it is now well known that they are intimately related to the structure of periodic geodesics. In this talk, we discuss analogous questions for Schrödinger operators, $-\Delta +V$ on $\mathbb R^d$, where $V$ is bounded together with all of its derivatives. Since the geodesic flow on $\mathbb{R}^d$ has no periodic trajectories (or indeed looping trajectories) one might guess that the spectral projector has a full asymptotic expansion. Indeed, for (quasi) periodic V this has been known since the work of Parnovski–Shterenberg in 2016. We show that when $d=1$, full asymptotic expansions continue to hold for any such $V$. When $d=2$, we give a large class of potentials whose spectral projectors have full asymptotics. Nevertheless, in $d \geq 2$, we construct examples where full asymptotics fail. Based on joint work with L. Parnovski and R. Shterenberg.

We discuss generalized norm resolvent convergence of discrete Schrödinger operators under several circumstances, including general (standard) lattices and the hexagonal lattice. We also discuss discrete Dirac operators, of which the definition itself is controversial. We explain several possible definitions and their continuum limits. These results are partially joint work with Y. Tadano (Tokyo Science University) and K. Mikami (Riken).

I will report on ongoing works with Yann Chaubet and Daniel Han-Kwan. We are interested in the long time dynamics of the nonlinear Vlasov equation on a negatively curved manifold when the interaction kernel is smooth. I will explain that, for small and smooth initial data carried away from the zero section, solutions to this nonlinear equation weakly scatter to an equilibrium state with an exponential rate. In our geometric framework, such scattering properties can be obtained by combining methods from kinetic theory with the microlocal tools developed for the study of Ruelle resonances and more specifically with the (normally hyperbolic) resolvent estimates of Nonnenmacher and Zworski.

In the first part of this talk, I will discuss recent results, joint with CL Fefferman and J Shapiro, on lower bounds for the magnetic hopping coefficient associated with a quantum particle in the 2D plane, confined by a double-well interacting with a perpendicular magnetic field. In the second part, I will explain how a non-uniform deformation of a non-magnetic honeycomb medium (e.g. a photonic crystal) induces effective magnetic and effective electric fields. Choosing a deformation corresponding to a constant perpendicular effective magnetic field gives rise to Landau-level (flat band) spectrum. I will present a continuum theory — tight binding is not applicable for photonic crystals — (joint work with J. Guglielmon and M. Rechtsman – Phys. Rev. A 103 2021 + work in progress), and then conclude with recent experimental confirmation of this theory (Barsukova et al. Nature Photonics 2024).

I will present new results on microwave experiments on open graphs. We measure the reflection of open microwave graphs, i.e., coaxial cables connected via T-junctions using vector network analyzer. From the reflection resonances can be extracted via the harmonic inversion techniques. Their dynamics on the degree of opening can be followed showing super radiant resonances if a balanced coupling to the environment is present, leading to the non-Weyl behavior. The open tetrahedral graph displays a rich parametric dynamic of the resonances in the complex plane presenting loops, regions of connected resonances and resonances approaching infinite imaginary parts. For Lasso graphs analytic results can be obtained showing the coexistence of bound states in the continuum (BICs) and superradiance.

This presentation introduces a new multi-frequency approach for reconstructing width defects in elastic waveguides. Unlike conventional inverse methods, our technique uses resonant frequencies known for their ill-conditioned propagation equations. By investigating the forward problem at these resonant frequencies, we employ a WKB approximation to characterize the wavefield for each modal component. Then, we apply this approximation to address the inverse problem, enabling stable reconstruction of width defects from partial wavefield measurements. We also provide numerical validations and comparative analyses against established methods, alongside experimental validations conducted at Institut Langevin, offering comprehensive insights into the efficacy and reliability of our approach.

We consider the derivative nonlinear Schrödinger equation on the circle and show that the corresponding Cauchy problem is globally well-posed in $H^1$ provided that the mass of initial data is strictly less than $8\pi$. This is a joint work with Hajer Bahouri.

The Laplace operator on the Euclidean space has no $L^2$ eigenfunctions, but by perturbing the metric or by adding a potential one can construct plenty of examples of operators with $L^2$ integrable eigenfunctions. The Landis conjecture states that any non-zero solution to $\Delta u + V u=0$ in the Euclidean space with real bounded $V$ cannot decay faster than exponentially near infinity.

If we are allowed to slightly perturb the coefficients of the Laplace operator (for instance taking a small smooth perturbation of the Euclidean metric and taking the Laplace operator for this metric) how fast can we force an eigenfunction of the perturbed Laplace operator to be localized?

We will review known results, related open questions and recent constructions of Nazarov and AL, Filonov and Krymskii, Pagano, AL and Krymskii of eigenfunctions to linear elliptic operators with smooth coefficients, which are localized much faster than exponentially.

In this talk I will discuss a recent approach to scattering theory and indicate some applications to the decay of waves on asymptotically flat, such as rotating Kerr black hole, spacetimes. Scattering theory in the spectral (time-independent) approach typically analyzes the near-spectrum behavior of the resolvent of a Laplacian-like operator on an asymptotically Euclidean space, including the existence of limiting resolvents in suitable function spaces. Here, in a geometric generalization of the setting, we set up a Fredholm theory on the spectrum, so that the spectral family is inverted directly as an operator, rather than having to take limits of the resolvent. The function spaces used are second microlocal, to precisely encode the behavior of outgoing spherical waves, but the approach captures not only Lagrangian regularity, but the full Fredholm picture by including also the analysis of a normal operator (operator valued symbol). This, together with the corresponding low energy analysis, then provides the analytic background for Hintz’s work on Price’s law (wave decay on black hole backgrounds) as well as the linearized black hole stability result of the author with Haefner and Hintz.

I will describe the asymptotic behavior of linear waves propagating on stationary and asymptotically flat spacetimes. I will focus on two settings: scattering by potentials on $\mathbb{R}^3$ with inverse cubic decay; second, a strengthening of Price's Law on Kerr spacetimes.

The geodesic flow on a closed Riemannian manifold with (strictly) negative curvature is the classical dynamics of a free particle and is very chaotic: it is Anosov, mixing, i.e. any smooth measure of probability on phase space evolving with the dynamics converges weakly towards the uniform (Liouville) measure, called equilibrium. In this talk we will describe the fluctuations around this equilibrium. We will explain that these fluctuations are governed by the wave equation on the manifold, i.e. the quantum dynamics. Techniques for the proofs are micro-local analysis, anisotropic Sobolev spaces and symplectic spinors. Collaboration with Masato Tsujii, arxiv:2102.11196.

The Hofstadter butterfly, a plot of the band spectra of almost Mathieu operators at rational frequencies, has become a pictorial symbol of the field of quasiperiodic operators and has gained renewed prominence through experimental study of moire materials. It is visually clear from this plot that for all irrational frequencies the spectrum must be a Cantor set, a statement that has been dubbed the ten martini problem. It has been established for the almost Mathieu operators, exploiiting various special features of this family. We will discuss a recently developed robust method allowing to establish it for a large class of one-frequency quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families. The proof builds on the recently developed concept of dual Lyapunov exponents and partial hyperbolicity of the dual cocycles.

I will review recent results on quantum transport in a periodic medium in the Boltzmann–Grad limit. There is an intriguing connection with fundamental problems that are of independent interest in number theory, including the distribution of quadratic Weyl sums and the pseudorandom features of certain lattice point distributions.

Topological insulators are striking quantum materials which block electricity in their interior but support robust currents along their boundary. The bulk-edge correspondence states that for topological insulators filling a half-space, the conductivity of the (straight) edge is equal to the Hall conductivity of the bulk. We present a generalization of this result to regions with arbitrary geometry. Specifically, we show that the conductivity of the edge(s) is the product of the Hall conductivity with a topological marker of the region. Joint work with Xiaowen Zhu.

In this talk, we will discuss some recent progress on the spectral statistics of compact negatively curved surfaces for the probabilistic model of random covers. In particular we will explain how by averaging over all possible covers it is possible to recover, in a large degree regime, for the smooth spectral counting functions, some of the spectral statistics of large random matrices models such as GOE and GUE.

I will explain a novel semiclassical calculus — called the Borel–Weil calculus — used to study (pseudo)differential operators defined over principal bundles.

This has two main applications: first, in the realm of dynamical systems, we prove that isometric extensions of Anosov flows are rapid mixing (that is, the correlation function decays faster than any polynomial power of time) under natural assumptions on the flow; in particular, this implies that the frame flow is rapid mixing as soon as it is ergodic. Secondly, we study the spectral theory of horizontal Laplacians induced by connections on principal bundles (these are sub-Riemannian Laplacians) and prove several properties (Weyl laws, quantum ergodicity, etc.).

My talk will mainly focus on this second application. More specifically, I will explain that horizontal Laplacians are (globally) hypoelliptic under a natural geometric assumption on the connection (it has dense holonomy group and the structure group of the bundle is semisimple). Surprisingly, this condition may be satisfied despite Hörmander's bracket condition not being verified at any single point (e.g. for flat bundles). Joint work with Mihajlo Cekić.

We show that the marked length spectrum of Riemannian surfaces with Anosov geodesic flow determines the metric up to isometry. Joint work with Lefeuvre and Paternain.

The study of twisted bilayer graphene is a topic of current interest in condensed matter physics: when two sheets of graphene are twisted by certain, coined magic, angles, the resulting material becomes superconducting. In this talk, we shall discuss a simple periodic Hamiltonian describing the chiral limit of twisted bilayer graphene, which displays some striking spectral properties occurring at magic angles. We show that the corresponding eigenfunctions decay exponentially in suitable geometrically determined regions as the angle of twisting decreases, which can be viewed as a form of semiclassical analytic hypoellipticity. This is joint work with Maciej Zworski.

A generalized product, refining the notion of a simplicial space, leads to an algebra of pseudodifferential operators quantizing an associated Lie algebroid. This construction includes many of the known pseudodifferential algebras on compact manifolds with corners and is intended as an organizing principle. For instance, every generalized product has a semiclassical extension.