In the last decade, there has been an increasing interest in the p-Laplacian, which plays an important role in geometry and partial differential equations. The p-Laplacian is a natural generalization of the Laplacian. Although the Laplacian has been much studied, not much is known about the nonlinear case p >1. Motivated by these facts, the purpose of the present paper is to review recent developments in the spectral theory of a specific class of quantum waveguides modeled by the Dirichlet Laplacian, i.e. p = 2, in unbounded tubes of uniform cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension. We discuss how the spectrum depends upon three geometric deformations: straightness, asymptotic straightness, and bending. Precisely, if the reference curve is straight or asymptotic straight, the essential spectrum is preserved. While dealing with bent tubes, such geometry produces a spectrum below the first eigenvalue. All the results confirm the literature for the Laplacian operator. The results are obtained via a very delicate analysis since the nonlinearity given by the p-Laplacian operator adds different types of difficulties with respect to the linear situation. These results are contained in a work written jointly with D. Krejčiřík.[-]

Hyperbolic (Anosov or Axiom A) flows have discrete Ruelle spectrum. For contact Anosov flows, e.g. geodesic flows, where a smooth contact one form is preserved, the trapped set is a smooth symplectic manifold, normally hyperbolic, and M. Tsujii, S. Nonnenmacher and M. Zworski, have given an estimate for the asymptotic spectral gap, i.e. that appears in the limit of high frequencies in the flow direction. We will propose a different approach that may improve this estimate. This will be presented on a simple toy model, partially expanding maps. Work with Tobias Weich.[-]

Reidemeister torsion was the first topological invariant that could distinguish between spaces which were homotopy equivalent but not homeomorphic. The Cheeger-Müller theorem established that the Reidemeister torsion of a closed manifold can be computed analytically. I will report on joint work with Frédéric Rochon and David Sher on finding a topological expression for the analytic torsion of a manifold with fibered cusp ends. Examples of these manifolds include most locally symmetric spaces of rank one. We establish our theorem by controlling the behavior of analytic torsion as a space degenerates to form hyperbolic cusp ends.[-]

I will discuss second order results for the length of nodal sets and the number of phase singularities associated with Gaussian random Laplace eigenfunctions, both on compact manifolds (the flat torus) and on subset of the plane. I will mainly focus on 'cancellation phenomena' for nodal variances in the high-frequency limit, with specific emphasis on central and non-central second order results.

Based on joint works with F. Dalmao, D. Marinucci, I. Nourdin, M. Rossi and I. Wigman.[-]

In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings.[-]

Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.[-]

Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.[-]

The Laplace–Beltrami operator in the curved Möbius strip is investigated in the limit when the width of the strip tends to zero. By establishing a norm-resolvent convergence, it is shown that spectral properties of the operator are approximated well by an unconventional flat model whose spectrum can be computed explicitly in terms of Mathieu functions. Contrary to the traditional flat Möbius strip, our effective model contains a geometric potential. A comparison of the three models is made and analytical results are accompanied by numerical computations.[-]

For any symmetric space $X$ of noncompact type, its quotients by torsion-free discrete isometry groups $\Gamma$ are locally symmetric spaces. One problem is to understand the geometry and analysis, especially the spectral theory, and interaction between them of such spaces. Two classes of infinite groups $\Gamma$ have been extensively studied:
$(1) \Gamma$ is a lattice, and hence $\Gamma$ $\backslash$ $X$ has finite volume.
$(2) X$ is of rank $1$, for example, when $X$ is the real hyperbolic space, $\Gamma$ is geometrically finite and $\Gamma$ $\backslash$ $X$ has infinite volume.
When $\Gamma$ is a nonuniform lattice in case $(1)$ or any group in case $(2)$, compactification of $\Gamma$ $\backslash$ $X$ and its boundary play an important role in the geometric scattering theory of $\Gamma$ $\backslash$ $X$. When $X$ is of rank at least $2$, quotients of $X$ of finite volume have also been extensively studied. There has been a lot of recent interest and work to understand quotients $\Gamma$ $\backslash$ $X$ of infinite volume. For example, there are some generalizations of convex cocompact groups, but no generalizations yet of geometrically finite groups. They are related to the notion of thin groups. One naturally expects that these locally symmetric spaces should have real analytic compactifications with corners (with codimension equal to the rank), and their boundary should also be used to parametrize the continuous spectrum and to understand the geometrically scattering theory. These compactifications also provide a natural class of manifolds with corners. In this talk, I will describe some questions, open problems and results.[-]