Nalini Anantharaman
Title: Spaces of hyperbolic surfaces and the Weil-Petersson model for a random surface
Abstract: The minicourse will begin with a gentle introduction to hyperbolic surfaces and their moduli spaces.
We will then describe the notion of random hyperbolic surface, mainly focussing on the probability measure defined by the Weil-Petersson measure.
We will describe some of the integration techniques found by M. Mirzakhani. We will focus on asymptotic phenomena in the “high genus” limit. Among others, we will show the limiting Poisson distribution
for the number of closed geodesics of lengths in [a, b], and explain the existence of a lower bound for the Cheeger constant.
1) Introduction of the model. McShane identity
2) McShane identity (continued). Mirzakhani’s integration formulas and consequences.
3) Poisson distribution for the number of closed geodesics. Cheeger constant. Survey of results on spectral gap for the laplacian.
Miklós Abert
Title: Graph convergence and manifolds
Abstract: I will introduce convergence of finite graphs and Riemannian manifolds and present some directions that naturally follow from using this language, including some open problems and (usually very) partial solutions to them.
Gregory Berkolaiko
Title: Universality of nodal count statistics in large graphs
Abstract: An eigenfunction of the Laplacian on a graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the first Betti number of the graph. We conjecture that the value distribution of the nodal surplus converges to Gaussian in any sequence of graphs of growing Betti number. There are partial analytical results and extensive numerical simulations for this conjecture in the setting of metric (“quantum”) graphs as well as some new conjectures for the discrete Laplacians. We will review the tools that have been useful so far — the nodal-magnetic theorem and an equivalence between a spectral average and a disorder average — as well as the current state of the conjecture.
Based on joint work with Lior Alon and Ram Band.
Benoit Collins
Title: Norm convergence for random unitary matrices
Abstract: We study the behavior of random representations of free groups. By random, we mean fixing compact matrix group and picking generators of the free groups iid according to the Haar measure. By behavior, mean specifically the limit of the operator norm of any non-commutative polynomial evaluated in the random generators, after picking an appropriate infinite sequence of matrix groups. The convergence of the norm towards the norm in a free probabilistic object (the reduced free group C*-algebra) is called strong asymptotic freeness. Strong asymptotic freeness was obtained in various cases, the unitary groups or the fundamental irreducible representation of the symmetric group. We will discuss why these results extend to a much broader class of matrix groups arising from sequences of representations of the unitary and orthogonal groups. This is based on joint works with Charles Bordenave.
Alix Deleporte
Title: Determinantal point projectors and spectral projectors for Schrödinger
Abstract: Determinantal point processes (DPPs) are a family of probabilistic models which represent the statistical properties of free fermions. Their study is further motivated by some natural mathematical examples of DPPs, such as random Hermitian matrices or random representations of finite groups.
To each (sequence of) orthogonal projectors on an space, one can associate a (sequence of) DPPs. This motivated us to study the semiclassical limit of usual spectral projectors in quantum mechanics from the point of view of their DPPs.
In this talk, I will present recent and on-going work with G. Lambert (UZH) on the asymptotics of spectral projectors for the semiclassical Schrödinger operators, and their applications to the convergence of DPPs at different scales.
Joel Friedman
Title: Trace Methods for Random Regular Graphs
Abstract: In this talk I present some simplifications in the trace method that allows for shorter proofs of eigenvalue bounds for random regular graphs of fixed degree and a large number of vertices. I will also explain how Shannon’s algorithm for variable-length codes plays a role in these methods. Part of this work is joint with David Kohler.
Jiaoyang Huang
Title: Extreme eigenvalues of random graphs with growing degrees
Abstract: I will discuss some results on extreme eigenvalue distributions of adjacency matrices of Erdős–Rényi graphs G(N,p) and random d-regular graphs , when the degree grows with the size N of the graph.
On the regime for Erdős–Rényi graphs and
for random d-regular graphs, we prove after proper normalization, the fluctuation of their extremal eigenvalues converges to the Tracy-Widom distribution. As a consequence, in the same regime of d, about 69% of all d-regular graphs have the second-largest eigenvalue strictly less than
. Our proof is based on constructing a higher order self-consistent equation for the Stieltjes transform of the empirical eigenvalue distributions. This is based on joint works with Horng-Tzer Yau.
Wolfgang König
Title: The parabolic Anderson model on a supercritial Galton–Watson tree
Abstract: We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model (PAM) on a supercritical Galton-Watson random tree with bounded degrees. This is the heat equation for the graph Laplacian with random potential, which we assume to be i.i.d. and doubly-exponentially distributed.
We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a subtree with minimal degree.
(joint work with Frank den Hollander and Renato dos Santos.)
Etienne Le Masson
Title: Quantum ergodicity for Eisenstein series on large genus random surfaces
Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces with cusps.
Joint work with Tuomas Sahlsten.
Michael Lipnowski
Title: The first non-zero Laplace eigenvalue of Weil-Petersson random surfaces of high genus: an introduction
Abstract: I’ll discuss joint work with Alex Wright proving that (Weil-Petersson) typical surfaces of large satisfy . The talk will be introductory in nature, surveying Selberg’s eigenvalue 1/4 conjecture, the Selberg trace formula, and its connection with Mirzakhani’s integration formulas.
Fabricio Macià
Title: Eigenstates of perturbed harmonic oscillators
Abstract: We present some results in the setting of the study of perturbations of Quantum Completely Integrable systems. In this talk we will focus on the model caso of the Quantum Harmonic Oscillator and study concentration properties of eigenstates in the presence of perturbations. Our results are formulated in terms of semiclassical microlocal defect measures; as it will turn out the structure of semiclassical measures, and the natures of the effect of the perturbation, strongly depends on the arithmetic properties of the set of characteristic frequencies of the non-perturbed Harmonic Oscillators. In some cases, it is possible to fully characterize the set of semiclassical measures for certain classes of quasi-modes. This is based on joint work with Víctor Arnaiz.
Michael Magee
Title: Near optimal spectral gaps for hyperbolic surfaces
Abstract: In this talk I’ll outline the proof that a uniform-random cover of a finite area non-compact hyperbolic surface has almost optimal relative spectral gap with probability tending to one as the degree of the cover tends to infinity.
This result is an analog of graph-theoretic results of Friedman and Bordenave-Collins, formerly conjectures of Alon and Friedman, respectively.
The proof involves combining the result of Bordenave-Collins on random permutations (in its fullest strength) with a parametrix construction of the resolvent on the random cover.
This is joint work with Will Hide.
Stéphane Nonnenmacher
Title: Eigenmode delocalization on Anosov surface
Abstract: The eigenmodes of the Laplacian on a smooth compact Riemannian manifold (M,g) can exhibit various localization properties in the high frequency limit, which depend on the dynamical properties of the geodesic flow. I focus on a “quantum chaotic” situation, namely assume that the geodesic flow is strongly chaotic (Anosov); this is the case if the sectional curvature of (M,g) is strictly negative. The Quantum Ergodicity theorem then states that almost all the eigenmodes become equidistributed on M in the the high frequency limit. The Quantum Unique Ergodicity conjecture claims that this behaviour admits no exception, namely all eigenstates should equidistribute in this limit.
This conjecture remaining inaccessible, a less ambitious goal is to constrain the possible localization of the eigenmodes. I will present some results on this question, in particular discuss a recent progress in the case of Anosov surfaces: all the eigenmodes must fully delocalize across all of M in the high frequency limit. More precisely, any semiclassical measure has full support.
The proof, which generalizes a previous work by Dyatlov-Jin in the constant curvature
case, uses the foliation of into stable and
unstable manifolds, and a Fractal Uncertainty Principle due to Bourgain-Dyatlov.
Hugo Parlier
Title: Ordering curves on hyperbolic surfaces
Abstract: This talk will be about different questions pertaining to the set of lengths of closed geodesics on hyperbolic surfaces. A classical question is to ask to what extent the set of lengths determine a surface. Here the setup will be slightly different: how much do you know about a surface if you know which curves are the shortest and without actually knowing their lengths?
Some of the results are joint with Cayo Dória.
Doron Puder
Title: The spectral gap of random covers of closed surfaces
Abstract: We study the spectral gap of a random covering space of a fixed surface, and show that for every ε>0, with high probability as the degree of the cover tends to ∞, the smallest new eigenvalue is at least 3/16-ε. Our main tool is a new method to analyze random permutations “sampled by surface groups”. This is based on joint works with Michael Magee and Frederic Naud.
Gabriel Rivière
Title: Poincaré series and linking of Legendrian knots
Abstract: On a compact surface of variable negative curvature, I will explain that the Poincaré series associated to the geodesic arcs joining two given points has a meromorphic continuation to the whole complex plane. This is achieved by using the spectral properties of the geodesic flow. Moreover, the value of Poincaré series at 0 can be expressed in terms of the genus of the surface after properly interpreting it in terms of the linking of two Legendrian knots. If time permits, I will explain how this result extends when one considers geodesic arcs orthogonal to two fixed closed geodesics.
This is a joint work with N.V. Dang.
Daniel Sánchez Mendoza
Title: Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box
Abstract: We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. For a special case in one dimension we give a complete proof.
Alex Wright
Title: The first non-zero Laplace eigenvalue of Weil-Petersson random surfaces of high genus
Abstract: I will discuss my joint work with Mike Lipnowski (arxiv.org/abs/2103.07496). I will build on the material presented in the talks of Lipnowski and Anantharaman, and focus on the typical geometry of closed geodesics at length scales that grow with the genus.