MAQD Conference Schedule

Abstract: It is about a joint work with Galina Perelman, where we investigate global well-posedness for the nonlinear derivative Schr\”odinger equation on the Torus. To our knowledge, the results to date, known on this subject, concern data with mass strictly less than $4\pi$. In this work, we show that global existence persists beyond this barrier.

Abstract: Random data theories for PDE’s were first developed by Bourgain in the 90’ in the context of his seminal works on Gibb’s measures for non linear Schr\”odinger equations. In Bourgain’s approach, randomness was an enemy because it forced to work at very rough regularity levels. It was only 15 years later that we realised with Tzvetkov that far from being an enemy, randomness could actually help in a PDE context and allow to exhibit examples where, while deterministically solutions to PDE’s exhibit bad behaviours, these pathological behaviours are (in some cases) actually quite rare and for suitable natural probability measures on the set of initial, they, almost surely, do not happen. I will present in this talk some basic ideas on this theory and some recent results (with their deterministic counterparts). This idea that randomness can help was actually also present in a series of works by Steve and collaborators and in some sense, the works I will present form a PDE counter part of some of his contributions.

Abstract: We consider Laplace eigenfunctions $-h^2 \Delta u = u$ in several settings, proving quantitative estimates on the $L^2$ mass of eigenfunctions in balls of radius depending on $h$. If $\mu(h) \to 0$ as $h \to 0$, we prove estimates of the form $\|u \|^2_{L^2(B_{\mu(h)})} = \O(\mu(h))$. In the case of a smooth manifold, we prove this up to scales $\mu \sim h$, and for interior neighborhoods in Euclidean domains, up to scale $\mu \ll h$. At a boundary point for Neumann eigenfunctions in piecewise smooth planar domains, we prove this up to scale $\mu \sim h^\delta$ for any $\delta<1$. The proofs are entirely stationary. This is joint work with John Toth.

Abstract: Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. In previous work with Jin and Nonnenmacher we showed that for Laplacian eigenfunctions on negatively curved surfaces, semiclassical measures have full support. This was restricted to dimension 2 because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and the speaker), was only known for subsets of the real line.

I will present several recent results on the support of semiclassical measures in higher dimensions, both on manifolds and in the toy model of quantum cat maps, contained in joint work with Jézéquel, joint work with Athreya and Miller, and work in progress by Kim. Some of these use the higher dimensional fractal uncertainty principle recently proved by Cohen. Others rely on separating the stable/unstable directions into fast and slow directions, and only applying the fractal uncertainty principle in the fast directions.

Abstract:
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. Using spherical mean operator, we will explain how 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.

Abstract: The aim of the talk is to present recent developments of high frequency analysis for sub-elliptic operators and in sub-Riemannian geometry. I will start with discussing why these questions are closely related to many aspects of harmonic analysis.

Abstract:
One of the most classical ways to numerically approximate the solution to high frequency scattering problems is the finite element method (FEM). In this method, one typically uses piecewise polynomials of some fixed degree p and a mesh-width h to approximate the solution. The fundamental questions is then: how should h be chosen (as a function of the frequency, k) so that the error in the numerical solution is small?

It has been known since the seminal work of Babuska and Ihlenberg that the natural conjecture of hk\ll 1 is not sufficient. Instead, one must require that (hk)^{2p} \rho(k)\ll 1 to maintain constant relative error, where \rho(k) is the norm of the relevant resolvent. In this talk, we will show that this condition can be substantially weakened by using a non-uniform mesh which takes advantage of the fact that errors are concentrated in some regions rather than others.

Abstract: On the 2 dimensional infinite cylinder, we define a measure on distributions $\phi$ that quantized
the massless sinh-Gordon action $S(\phi)=\int_C (|d\phi|^2+\cosh(\phi))$. This is done through spectral theory
and probabilistic means, with a Markov dynamics underlying the construction.
The associated propagator has a generator with discrete spectrum and one can use it to define
correlation functions for this model. This is joint work with Gunaratnam and Vargas.

Abstract: In this talk I will present a natural way of approximating a
subRiemannian structure by a sequence of Riemannian metrics.
I will focus on how a natural volume arises in the process and whether
the sequence of associated Laplace eigenvalues converges.

This is joint work with Mohammad Hussein Harakeh.

Abstract: We study properties of geodesics that are orthogonal to two
convex subsets of the flat torus. We define Poincaré series associated
with the set of lengths of these curves. Depending on the regularity of
the convex sets, we prove continuation of Poincaré series to different
regions of the complex plane, and relate their singularities to the
geometry of the convex sets and the spectrum of an associated elliptic
operator.
This is joint work with Yannick Guedes Bonthonneau, Nguyen Viet Dang and
Gabriel Rivière.

Abstract: An isometric extension of an Anosov flow is an extension to a G-principal bundle over the base space, where G is a compact Lie group. A typical example if provided by the frame flow over a negatively-curved Riemannian manifold. In this talk, I will explain a recent result obtained in collaboration with M. Cekić, showing that isometric extensions exhibit rapid mixing under a natural assumption (i.e., the decay of correlations is faster than any polynomial power of time). In this problem, there is a dichotomy based on whether G is Abelian or semisimple. Interestingly, the semisimple case turns out to be easier due to a Diophantine property satisfied by semisimple Lie groups. Therefore, I will mainly focus on the Abelian case.

Abstract: We discuss recent results on the essential self-adjointness of
Klein-Gordon type operators on several classes of spacetimes. The
first one is the asymptotically flat spacetime, which was studied
previously by A. Vasy (J. Spectral Theory 2020) and by us (Ann. H.
Lebesgue 2021), but we present a new simpler proof (Ann. H. Poincaré
2023). We also discuss the essential self-adjointness for the
asymptotically static spacetime, which is Cauchy compact (Comm. Math.
Phys. 2023). These results are joint work with Kouichi Taira (
Ritsumeikan University).

In certain phase spaces (Euclidean space, projective spaces, tori),
quantum states can
be represented by holomorphic functions (or sections) defined on the
phase space. I will speak of the “Bargmann function” or the quantum state.

I plan to recall various (relatively old) results on the nodal sets of these holomorphic
functions, a topic which was the focus of Steve Zelditch’s interest
during many years.
I will focus on the simplest case of 1D systems, where the
nodal set is a discrete set of zeroes. This set of points essentially contains all
the information on the quantum state, and it was baptized the “stellar
representation” of the quantum state by Leboeuf-Voros.

One question concerns the distribution of this zero set, for
various classes of quantum states, in particular eigenstates of quantum
Hamiltonians, or quantum maps, in the semiclassical regime. In
particular, a clear difference occured between eigenstates of a
Hamiltonian (which is, in 1D, a Liouville- integrable system), and of
a quantized chaotic map. For the latter, a consequence of quantum
ergodicity is the asymptotic equidistribution of the zero set over the
phase space.

I will also address the dynamics of the zero set, when a quantum
Hamiltonian flow applies to the quantum state. In this case, the only
rigorous results concern the class of Gaussian random states, for
which one finds a connection with the classical dynamics.

Those are Joint results with André Voros.

Abstract: I will discuss asymptotic properties of eigenfunctions for
sub-Riemannian Laplacians associated with a contact structure in
dimension 3. More precisely, I will discuss their asymptotic invariance
properties and explain how these are modified when the operator is
perturbed by a subprincipal symbol. This is a joint work with Victor
Arnaiz (Bordeaux).

Abstract: We shall explore the role that curvature plays in harmonic analysis on compact manifolds. We shall focus on estimates that measure the concentration of eigenfunctions. Using them we are able to affirm the classical Bohr correspondence principle and obtain a new classification of compact space forms in terms of the growth rates of various norms of (approximate) eigenfunctions.

This is joint work with Xiaoqi Huang following earlier work with Matthew Blair.

Abstract: I will describe the covariant setup for spectral theory on any stationary globally hyperbolic spacetime. Local spectral invariants that are analogous to the heat invariants in the Riemannian setting exist also in this setting, but are more complicated. These invariants have been obtained in collaboration with Steve Zelditch as part of a wider program. I will summarise some of the results obtained as part of that.

Abstract: A central question of Euclidean harmonic analysis is; when does a multiplier

$$Mf=\mathcal{F}^{-1}\left[m(\cdot)\mathcal{F}[f](\cdot)\right]$$

defined as an operator $L^{2}\to L^{2}$ extend to a bounded operator $L^{p}\to L^{q}$? The Bochner-Riesz multipliers where

$$m_{R}(\xi)=\left(1-\frac{|\xi|^{2}}{R^{2}}\right)_{+}^{\delta}$$

are one well-known example of these type of operators. On manifolds we can consider analogous questions about whether spectral multipliers $F(\sqrt{\Delta})$ are bounded. Such questions have been long known to be connected to the growth properties of quasimodes (approximate solutions) to the eigenfunction equation $\sqrt{\Delta}u=\lambda u$. In this talk we will see how we can formalise the relationship between growth properties of quasimodes and boundedness of spectral multipliers and use the relationship to obtain new results about the latter.

Let $(M,g)$ be a $C^{\infty}$ compact, Riemannian manifold and $u_h \in C^{\infty}(M)$ be a sequence of $L^2$-normalized Laplace eigenfunctions with $(-h^2 \Delta_g – 1) u_h = 0$. Let $H \subset M$ be a smooth hypersurface.
In the terminology of Zelditch and Toth, the hypersurface $H$ is {\em good} relative to the sequence $u_h$ if there exist constants $h_0(H), C_H >0$ such that for all $h \in (0,h_0(H)],$

$$ \int_H |u_h|^2 d\sigma_H \geq e^{ – C_H/h}.$$

In the talk, I will describe some recent results (joint with Yaiza Canzani) on goodness of hypersurfaces relative to eigenfunction sequences and give some applications to nodal sets of eigenfunction sequences.

Abstract: I will discuss two examples of second microlocalization: one in the semiclassical setting and one in scattering theory. Both allow a simple analysis with interesting conclusions.

Abstract: If you know the shape of a drum head, can you determine where it has been struck, up to symmetry, by ear alone? This problem echoes the celebrated problem of Kac ’66, “Can one hear the shape of a drum?” except that what we hear is the pointwise Weyl counting function instead of the Laplace-Beltrami spectrum.

In this talk, I will define the problem, run a few examples, and go over a few results, the core of which is that one can hear where `most’ drums are struck.

The results in this talk come out of joint work with Yakun Xi, Feng Wang, and Shi-Lei Kong. This problem is dedicated to Steve Zelditch. We like to imagine that he’d have liked this problem, and we know for certain he would have had a lot of insight to share about it.

Abstract: In joint work with Jeff Galkowski, we show that for the
Lindblad evolution defined using (at most) quadratically growing
classical Hamiltonians and (at most) linearly growing classical jump
functions (quantized into jump operators assumed to satisfy certain
ellipticity conditions and modeling interaction with a larger system),
the evolution of a quantum observable remains close to the classical
Fokker–Planck evolution in the Hilbert–Schmidt norm for times vastly
exceeding the Ehrenfest time (the limit of such agreement with no jump
operators). The time scale is the same as in the recent papers by
Hernandez–Ranard–Riedel but the statement and methods are different.
The results are illustrated by numerical experiments due to Zhen
Huang.