Abhishek Saha

Monday July 23, 2:00-2:50

Title: Sup-norms of Maass forms in the level aspect for compact arithmetic surfaces

Abstract: The asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds plays an important role in various areas of mathematics, including number theory, quantum chaos, and geometry. A natural question here is to prove non-trivial upper bounds on the supremum norm of such eigenfunctions; this is known as the sup-norm problem. Beginning with the pioneering work of Iwaniec-Sarnak from 1995, there has been a lot of work on this problem in the large eigenvalue limit (where the manifold is kept fixed). However, it is also very interesting to consider the case when the manifold itself varies, in particular by taking quotients of a (fixed) symmetric space with subgroups whose co-volume goes to infinity. Restricting ourselves to the arithmetic case (where one can exploit the action of Hecke operators), this gives rise to the problem of understanding the asymptotics of automorphic forms in the level aspect.

The rank 1 case of this problem relates to Maass forms on the upper-half plane that are invariant under the action of (or more generally, transform by unitary characters of) varying discrete arithmetic subgroups. In recent years, there has been a fair bit of progress in the special case of classical Maass newforms (the arithmetic subgroup here is Gamma_0(N), which is not co-compact). I will begin by reviewing some of my recent results in this area. However, the main focus of this talk will be the compact case, where I will consider Maass forms transforming with respect to (unitary characters of) unit groups of orders of an indefinite quaternion division algebra over Q. I will present an upper bound for the sup-norm in the level aspect that is valid for arbitrary orders and improves upon the trivial bound in almost all cases. Despite the generality of the setup (and the lack of assumptions related to newform theory), our bound is sufficiently strong that (when restricted to specific cases of interest) it improves upon the "local bound" for several families of automorphic forms of non-squarefree level.

Henrik Uebershaer

Monday July 23, 3:00-3:50

Title: Scarred quasimodes on rational polygons

Abstract: An important question in quantum chaos concerns the distribution of eigenfunctions of the Laplacian on a compact Riemannian manifold in the limit as the eigenvalue tends to infinity. The quantum ergodicity theorem states that on manifolds, whose geodesic flow is ergodic with respect to Liouville measure, a full density subsequence of eigenfunctions become equidistributed in phase space. A major open problem concerns the existence of sparse subsequences that localise on periodic orbits -- also known as "scars".

In this talk, I will deal with pseudo-integrable billiards -- examples of which are Seba's billiard and rational polygons. Such billiards are classically integrable, yet show features of chaos at the quantum level. In particular, I will report on the construction of quasi-modes of the Laplacian on rational polygons which become scarred along a finite number of vectors in momentum space. This is joint work in progress with Omer Friedland at Jussieu.

Igor Wigman

Monday July 23, 4:30-5:20

Title: Planck scale mass distribution of toral Laplace eigenfunctions.

Abstract: The presented research is based on works joint with A. Granville and N. Yesha.

We study the small scale distribution of the L2-mass of eigenfunctions of the Laplacian on the two and three dimensional flat tori. For 2d we show that equidistribution holds down to a small power of log above Planck scale, and also that the L2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale, hence refining a recent result of Lester-Rudnick in this case.Furthermore, In 2d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In 3d the asymptotic behaviour of the variance is analysed in a more restrictive scenario ("Bourgain's eigenfunctions").

The said results rest on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.

Zeev Rudnick

Tuesday July 24 10:00-10:50

Title: Points on nodal lines with given direction

Abstract: We study the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves. Joint work with Igor Wigman.

Peter Humphries

Tuesday July 24 11:00-11:50

Title: $L^2$-Restriction, Small Scale Equidistribution, Period Formulae, and Subconvexity

Abstract: I will discuss several $L^2$-restriction problems for automorphic forms and its similarity with small scale equidistribution. I will highlight in each case how the relation between these problems and period formulae for automorphic forms, and the subconvexity results required to prove nontrivial bounds.

Junehyuk Jung

Tuesday July 24, 2:00-2:50

Title: Ergodicity and the number of nodal domains of Laplacian eigenfunctions

Abstract: The growth of the number of nodal domains of Laplacian eigenfunctions on a manifold is closely related to the ergodicity of the geodesic flow. For instance, on a surface with an isometric involution whose geodesic flow is ergodic (e.g. arithmetic hyperbolic triangles), typical Laplacian eigenfunctions have growing number of nodal domains. In this talk, I'm going to talk about circle bundles over a closed surface equipped with Kaluza--Klein metrics (e.g. compact quotients of $SL(2,\mathbb{R})$), for which I prove that typical eigenfunctions have only two nodal domains. Note that for such examples, geodesic flow is never ergodic. I also will present an explicit orthonormal eigenbasis on the 3 torus where all of them have only two nodal domains. These are based on joint works with Seung Uk Jang and Steve Zelditch.

Bingrong Huang

Tuesday July 24, 3:00-3:50

Title: Sup-norm and nodal domains of CM Maass forms.

Abstract: I will talk about the sup-norm bound and the lower bound of the number of nodal domains for CM Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $\phi$ be a CM Maass form with spectral parameter $t_{\phi}$, then we prove that $\|\phi\|_{\infty} \ll t_{\phi}^{3/8+\varepsilon} \|\phi\|_2$, which is an improvement over the bound with exponent 5/12 given by Iwaniec and Sarnak. We also prove that the number of nodal domains grows faster than $t_{\phi}^{1/8-\varepsilon}$ for almost all CM Maass forms.

Paul Nelson

Tuesday July 24, 4:30-5:20

Title: Quantum variance of compact arithmetic surfaces

Abstract: I will talk about the quantum variance of Hecke--Maass cusp forms on arithmetic hyperbolic surfaces. In the case of the non-compact modular surface, the asymptotic quantum variance was determined in work of Luo--Sarnak, Zhao and Sarnak--Zhao, using a method specific to that case. I will describe analogous results in the compact case, obtained by a different method based on the theta correspondence. This will involve some discussion of constructing microlocal lifts via convolution operators, identities and asymptotics involving integrals of theta functions, the Rallis inner product formula, and the method of coadjoint orbits.