Etienne Le Masson, Eigenfunctions on random hyperbolic surfaces
High frequency eigenfunctions on hyperbolic surfaces are known to exhibit some universal behaviour of delocalisation and randomness. We will introduce some results on the behaviour of eigenfunctions on random compact hyperbolic surfaces, in the limit where the genus (or equivalently the volume) tends to infinity, and the frequency is in a fixed window. These results suggest that in this large scale limit we can expect similar universal behaviour. We will focus on the Weil-Petersson model of random surfaces introduced by Mirzakhani.
Based on joint works with Tuomas Sahlsten and Joe Thomas.
Bart Michels, Mean square asymptotics and oscillatory integrals for maximal flat submanifolds of locally symmetric spaces
Given a compact locally symmetric space of non-compact type, we present a mean square asymptotic for integrals of eigenfunctions along maximal flat submanifolds, constrained to eigenfunctions with suitably generic spectral parameter. This is motivated by questions concerning the maximal size of automorphic periods. The proof uses the pre-trace formula. The analysis of orbital integrals requires knowledge about the geometry of maximal flat submanifolds of the globally symmetric space S. When S is the hyperbolic plane, modeled by the upper half plane, the maximal flat submanifolds are geodesics, and they are lines or half-circles orthogonal to the real axis. The midpoints of the half-circles play a critical role, as do their analogues in higher rank spaces, and one is led to generalize their properties as well as other facts about maximal flat submanifolds.
Matthew Blair, Lp bounds for eigenfunctions at the critical exponent
We consider upper bounds on the growth of L^p norms of eigenfunctions of the Laplacian on a compact Riemannian manifold in the high frequency limit. In particular, we seek to identify geometric or dynamical conditions on the manifold which yield improvements on the universal L^p bounds of C. Sogge. The emphasis will be on bounds at the "critical exponent", where a spectrum of scenarios for phase space concentration must be considered. We then discuss a recent work with C. Sogge which shows that when the sectional curvatures are nonpositive, there is a logarithmic type gain in the known L^p bounds at the critical exponent.
Yaiza Canzani, Eigenfunction concentration via geodesic beams
A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of Laplace eigenfunctions. Because of this, it is crucial to understand how various measures of eigenfunction concentration respond to the background dynamics of the geodesic flow. In collaboration with J. Galkowski, we developed a framework to approach this problem that hinges on decomposing eigenfunctions into geodesic beams. In this talk, I will present these techniques and explain how to use them to obtain quantitative improvements on the standard estimates for the eigenfunction's pointwise behavior, Lp norms, and Weyl Laws. One consequence of this method is a quantitatively improved Weyl Law for the eigenvalue counting function on all product manifolds.
Emmett Wyman, Eigenfunctions restricted to submanifolds and their Fourier coefficients
Consider a Laplace-Beltrami eigenfunction on some compact manifold, and restrict it to a compact submanifold. We may write the restricted eigenfunction as a combination of eigenbasis elements intrinsic to the submanifold, whose coefficients we will call Fourier coefficients. What does the spectral decomposition of the restricted eigenfunction look like? How much of the mass of the Fourier coefficients is concentrated near the eigenvalue? Do the Fourier coefficients "feel" the geometry of the submanifold or ambient manifold? If so, how?
I will present joint work with Yakun Xi and Steve Zelditch on such questions. Indeed, various aspects of these Fourier coefficients reflect the geometry of the submanifold and ambient space. Of particular importance are configurations of "geodesic bi-angles," which consist of a pair of geodesics, one in the ambient manifold and one intrinsic to the submanifold, with shared endpoints. These bi-angles arise in the wavefront set analysis a la the Duistermaat-Guillemin theorem.
Angela Pasquale, Resonances of the Laplacian on Riemannian symmetric spaces of the noncompact type of rank 2
Let X=G/K be a Riemannian symmetric space of non-compact type and let Delta be the positive Laplacian of X, with spectrum \sigma(\Delta). Then the resolvent R(z)=(\Delta-z)^{-1} is a holomorphic function on \mathbb{C}\setminus \sigma(\Delta) with values in the space of bounded linear operators on L^2(X). If R admits a meromorphic continuation across \sigma(\Delta), then the poles of the meromorphically extended resolvent are called the resonances of \Delta. At present, there are no general results on the existence and the nature of resonances on a general X=G/$. In this talk, we will mostly focus on the case of rank two.
This is part of a joint project with J. Hilgert (Paderborn University) and T. Przebinda (University of Oklahoma).
Melissa Tacy, Applications of semiclassical analysis to harmonic analysis
Semiclassical analysis is a form of microlocal analysis specialised to study parameter problems. It is highly effective for treating "high frequency/energy" style problems arising in harmonic analysis. In this talk I will discuss some of the ideas, heuristics and techniques of semiclassical analysis with a particular focus on applications in harmonic analysis.
Yunfeng Zhang, Fourier restriction bounds on compact symmetric spaces
In this talk I will make a survey of bounds of "Fourier restriction" type on compact Lie groups and more generally compact globally symmetric spaces. These include Laplace-Beltrami eigenfunction bound, Strichartz estimate for the Schr\"odinger equation, and joint eigenfunction bound for invariant differential operators. Optimal bounds are all open, for which a more refined combination of Lie theory and analysis would be needed.
Jean-Philippe Anker, Dispersive PDE on noncompact symmetric spaces
My talk will be devoted to the Schrödinger equation and to the wave equation on general Riemannian symmetric spaces of noncompact type. The main issue consists in obtaining good pointwise estimates of their fundamental solutions. This is achieved by combining the inverse spherical Fourier transform with the following tools : on the one hand, a barycentric decomposition, which allows us to handle the Plancherel density as if it were a differentiable symbol, and, on the other hand, an improved Hadamard parametrix for the wave equation. As consequences, we deduce dispersive estimates and Strichartz inequalities for the linear equations, which are stronger than their Euclidean counterparts, as well as better results for the nonlinear equations. All this is based on joint works including several collaborators : Vittoria Pierfelice in rank one, Hong-Wei Zhang in higher rank, with contributions by Maria Vallarino and Stefano Meda.
Jasmin Matz, Quantum ergodicity in the level aspect
A classical result of Shnirelman and others shows that closed Riemannian manifolds of negative curvature are quantum ergodic, meaning that on average the probability measures |f|^2 dx on M, with f running through normalized Laplace eigenfunctions on M with growing eigenvalue, converge towards the Riemannian measure dx on M.
Following ideas of Abert, Bergeron, Le Masson, and Sahlsten, we look at a related situation: We want to consider certain sequences of manifolds together with Laplace eigenfunctions of approximately the same eigenvalue instead of high energy eigenfunctions on a fixed manifold. In my talk I want to discuss joint work with F. Brumley in which we study this situation in higher rank for sequences of compact quotients of SL(n,R)/SO(n).