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21-22 Mar 2019 Paris (France)
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Titles and abstractsSpeaker: Miklos Abert Title: Eigenfunctions and Random Waves in the Benjamini-Schramm limit
Abstract: We investigate the asymptotic behavior of eigenfunctions on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. This leads to a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and at the same time formulates a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.
Speaker: Matthew de Courcy-Ireland Title: Kesten-McKay law for the Markoff surface mod p
Abstract: The Markoff surface is a cubic surface with the special feature that it is only quadratic in each variable separately. The Markoff surface modulo a prime number forms a 3-regular graph where the edges correspond to exchanging the two roots of such a quadratic. We show that for increasingly large primes, the eigenvalues of these graphs asymptotically follow the same law as the eigenvalues of a random 3-regular graph. The proof is based on the method of moments and takes advantage of an action of GL(2,Z) on the Markoff surface. This is joint work with Michael Magee.
Speaker: Alix Deleporte
Title: Eigenfunctions on random surfaces
Abstract: In a recent result by Abert-Bergeron-Le Masson, the authors study random superpositions of Laplace eigenfunctions on hyperbolic surfaces. If a sequence of surfaces converges (in the sense of Benjamini-Schramm) to the hyperbolic plane, then a random superposition of eigenfunctions in a shrinking energy window behaves like a Gaussian field (the monochromatic hyperbolic Gaussian field). Benjamini-Schramm convergence implies that the genus of the surface tends to infinity, yielding the following question: what does a typical eigenfunction on a typical surface of large genus looks like? There is a natural probability measure on the moduli space of hyperbolic surfaces (Weil-Petersson), from which one can formulate an analogue of Berry's conjecture: the result of Abert-Bergeron-Le Masson would be true for single, randomly chosen, eigenfunctions (without a need to take a random linear combination of them). In this talk, I will play the Devil's advocate and present a few strong hints towards failure of convergence to a Gaussian field. Speaker: Mikolaj Fraczyk
Title: Benjamini-Schramm sampling of eigenfunction at intermediate scales.
Abstract: I will report on an ongoing project with Miklos Abert where we investigate the asymptotic properties of eigenfunction of Laplacian on a manifold M rescaled by h^{-\alpha} where the exponent \alpha is between 0 and 1 and h is the Planck scale. Like in the usual quantum ergodicity question we can look at the limits of semiclassical measures associated to such sequences. Using diophantine methods we establish some regularity properties of such intermediate semiclassical limits when M is a flat torus. This extends some of the results of Dmitry Jakobson on the quantum limits on flat tori.
Speaker: Étienne Le Masson Title: Quantum ergodicity on large genus hyperbolic surfaces.
Abstract: We will present recent results concerning the delocalisation of eigenfunctions of the Laplacian on hyperbolic surfaces of large genus (and more generally hyperbolic spaces of large volume). These results are stated in the general setting of Benjamini-Schramm convergence and we will try to connect them on one hand to the quantum unique ergodicity theory for Maass forms in the level aspect, and on the other hand to the study of random surfaces of large genus. The results are weaker than what we could expect in these settings, but are more general and deterministic. The proof is based on ergodic theory methods. Based on joint works with Miklos Abert, Nicolas Bergeron and Tuomas Sahlsten. Speaker: Simon Marshall
Title: On the subconvexity problem for U(n) x U(n+1)
Abstract: I will discuss recent work of mine on the subconvexity problem for U(n) x U(n+1). This work proceeds via period integrals, using work of Beuzart-Plessis and Zhang on the unitary Ichino-Ikeda conjecture, and attempts to bound the relevant period using arithmetic amplification and the microlocal lift vectors of Nelson-Venkatesh. Speaker: Jasmin Matz
Title: Asymptotics of traces of Hecke operators
Abstract: The distribution of spectral parameters in various families of automorphic representations has many applications to families of automorphic L-functions, e.g. low-lying zeros. I want to talk about joint work with T. Finis in which we prove an effective equidistribution result for Satake parameters of unramified automorphic forms on many split reductive groups. Compared to previously known results for GL(n), we can improve the bounds for the remainder terms. This also yields improved bounds for the remainder of the Weyl law for the cuspidal spectrum for several locally symmetric spaces.
Speaker: Bart Michels
Title: Lower bounds for geodesic periods on hyperbolic surfaces
Abstract: On compact arithmetic hyperbolic surfaces, Iwaniec and Sarnak have shown that there exist sequences of Laplacian eigenfunctions whose sup norms grow with the eigenvalue. The lower bound they found can be explained by the random wave model. In fact, it is consistent with the analogue of the law of the iterated logarithm in this context. Milićević later proved the existence of sequences of eigenfunctions whose sup norms grow much faster than what can be explained by randomness. In both cases, large values are found at arithmetically charged points. In this talk, I will exhibit eigenfunctions on compact arithmetic surfaces, whose periods along a closed geodesic grow nontrivially, relative to the average value of such periods. The lower bound stands at the level of what is predicted by the random wave model.
Speaker: Paul Nelson
Title: Applications of the orbit method to the analysis of automorphic forms
Abstract: I will discuss joint work with Akshay Venkatesh in which we use microlocalized test vectors (inspired by the orbit method) and Ratner theory to study mean values of L-functions on Gross--Prasad pairs. I will also indicate some further applications of these methods.
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