Speakers  Affiliation  Titles and Abstracts 
Anshul Adve  Princeton University  A spectral gap for spinors on hyperbolic surfaces 
[Abstract]
I'll describe a sequence of hyperbolic surfaces with a choice of spin structure and with genus going to infinity, such that the bottom eigenvalue of the Laplacian on spinors is uniformly bounded below. The surfaces can be taken to be arithmetic. A fun feature of the construction is that it makes use of four different perspectives on Riemann surfaces: algebraic, complex analytic, geometric, and arithmetic. Along the way we'll see that when the Riemann surface in question is sufficiently symmetric, one can pass between these perspectives in an explicit and fruitful manner. This is joint with Vikram Giri.


Mikhail Belolipetsky  IMPA  Geometric expanders 
[Abstract]
In a joint work with Hannah Albert we showed that closed arithmetic hyperbolic 3dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1skeletons are harder and harder to embed nicely in Euclidean space. This is achieved by generalizing an inequality of Gromov and Guth to hyperbolic norbifolds and finding nearly optimal geodesic triangulations of arithmetic hyperbolic 3orbifolds. The relation between the euclidean embeddings and expander graphs goes back to the pioneering work of Kolmogorov and Barzdin from the 1960s. In the talk I will discuss these results with an emphasis on some related open problems.


Sean Eberhard  Queen's University Belfast  Babai's conjecture for classical groups with random generators, part 2 
[Abstract]
Consider G = SL_n(q) acting on V \ {0}, where V = F_q^n is the defining module, and consider the Schreier graph defined by k random elements x_1, ..., x_k \in G. In this talk we will show that the there is a spectral gap with high probability provided that k is larger than a constant depending only on q. More precisely we will show that the second largest eigenvalue of the normalized adjacency matrix is bounded by (1+o(1)) q sqrt(2k1)/k. The argument is based on the classical trace method approach of Broder and Shamir (1987), suitably adapted to the linear setting. Analogous results hold for other classical groups and for the action on rtuples of vectors for bounded r, which is a linear analogue of a result for S_n of Friedman, Joux, Roichman, Stern, and Tillich (1998). Time permitting we will also outline these generalizations. As an application we prove Babai's conjecture on the diameter of simple classical groups for generating sets which include both a boundeddegree element and sufficiently many random generators. Joint work with Urban Jezernik.


Sam Edwards  Durham University  The bottom of the L^2 spectrum of higherrank locally symmetric spaces 
[Abstract]
For a rank one geometrically finite locally symmetric space Γ\X, the bottom of the L^2 spectrum of the Laplace operator is a simple eigenvalue corresponding to a positive eigenfunction if and only if the critical exponent of Γ is strictly greater than half the volume entropy of X. In particular, there exist infinite volume rank one locally symmetric spaces with square integrable positive Laplace eigenfunctions. In contrast, a higherrank symmetric space Γ\X without rank one factors has a square integrable positive Laplace eigenfunction if and only Γ is a lattice. We will explain some aspects of the connection between square integrability of positive Laplace eigenfunctions and PattersonSullivan and BowenMargulisSullivan measures in the higherrank setting. Based on joint work with Oh and FraczykLeeOh.


Alex Gamburd  CUNY Graduate Center  Strong Approximation for Varieties of Markoff Type 
[Abstract]
The Markoff equation \( x^2+y^2+z^2=3xyz \), which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.


Will Hide  Durham University  Spectral gaps for random covers of hyperbolic surfaces 
[Abstract]
Based on joint work with Michael Magee.


Urban Jezernik  University of Ljubljana  Babai's conjecture for classical groups with random generators, part 1 
[Abstract]
The diameter of a finite group G equipped with a generating set S is the smallest number k so that every element of G can be written as a product of at most k elements from S. Babai's conjecture predicts that diameters of finite simple groups should be polylogarithmic in the size of the group. Much is known about this conjecture, yet it is still open for the family of groups PSL_n(F_q) as n tends to infinity and q is fixed. We investigate this family in the generic situation when the generating sets consist of random elements and prove that Babai's conjecture holds in this case. Our argument naturally splits into two parts: first we show how to quickly get a transvection using random elements, and second we use spectral gaps to quickly get all other transvections and complete the proof. In this talk, we will gently introduce the problem, outline the strategy of our argument for PSL_n(F_q), and give some details on the first part of the proof. Joint work with Sean Eberhard.


Petr Kravchuk  King's College London  Spectral gaps and Conformal bootstrap 
[Abstract]
In this talk I will review an approach to deriving upper bounds on the gap in the Laplace spectrum of hyperbolic 2orbifolds, inspired by numerical bootstrap methods from conformal field theory. As I will explain, these bounds are obtained from consistency conditions for integrals of quadruple products of holomorphic differentials and their derivatives, and can be numerically very close to what is believed to be the largest gap. This might suggest that the bounds can be improved to be tight, but I will provide strong evidence that this is impossible. Based on work with Dalimil Mazac, Sridip Pal and Alexander Radcliffe.


Ursula Hamenstadt  University of Bonn  Spectral gap and eigenfunctions on hyperbolic 3manifolds 
[Abstract]
We summarize what is known about the spectral gap and
properties of the corresponding eigenfunctions for (possibly not closed)
hyperbolic manifolds. We also develop a rather speculative picture on
the relation between spectral gap, volume and Heegaard genus as it arises
in progress on effective hyperbolization.
This is in part joint work with Gabriele Viaggi, and joint work with
Elia Fioravanti, Frieder Jaeckel and Yongquan Zhang.


Dalimil Mazac  IAS Princeton  Spectral bounds on hyperbolic 3manifolds from the associativity of multiplication 
[Abstract]
I will discuss bounds on the lowenergy spectra of Laplacians on compact hyperbolic 3manifolds and orbifolds. The key ingredient is an infinite collection of identities satisfied by 1) the spectra of the Laplacians on powers of the cotangent bundle, and 2) the integrated triple products of the corresponding eigenfunctions. These spectral identities follow from the consistency of associativity of multiplication of functions on $\Gamma \backslash \mathrm{PSL}_2(\mathbb{C})$ with the spectral decomposition. They can be turned into spectral bounds via linear programming. I will compare the bounds with the spectra of various hyperbolic 3manifolds and show that they are often nearly sharp. Finally, I will explain how this approach is related to the conformal bootstrap program in conformal field theory, which served as the original inspiration. Based on work with J. Bonifacio, P. Kravchuk and S. Pal.


Wenyu Pan  University of Toronto  Exponential mixing of frame flows for geometrically finite hyperbolic manifolds 
[Abstract]
The frame bundle of an ndimensional hyperbolic manifold X is the homogeneous space Γ\SO(n, 1)° for some discrete subgroup Γ and the frame flow is given by the right translation action by a oneparameter diagonalizable subgroup. We assume that Γ is Zariski dense and X is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. We endow the frame bundle with the unique probability measure of maximal entropy called the BowenMargulisSullivan measure. In joint work with Jialun Li and Pratyush Sarkar, we prove that the frame flow is exponentially mixing. The proof uses a countably infinite coding and Dolgopyat's method à la SarkarWinter and TsujiiZhang. To overcome the difficulty in applying Dolgopyat's method due to the cusps of nonmaximal rank, we prove a uniform large deviation property for symbolic recurrence to certain large subsets of the limit set of Γ.


Ori Parzanchevski  Hebrew University of Jerusalem  Random walks on Ramanujan graphs and complexes 
[Abstract]
Recent works of LubetzkyPeres, Sardari and NestoridiSarnak have shown that nonbacktracking random walks on Ramanujan graphs exhibit optimal cutoff; namely, the L^1mixing time of the walk is around log_k(N) steps (for a kregular walk on N vertices). I will explain these results, their generalization to Ramanujan complexes of arbitrary dimension, and consequences for optimal expansion in finite simple groups.


Bram Petri  IMJ and Sorbonne University  Linear programming bounds for hyperbolic surfaces 
[Abstract]
I will speak about extremal problems in the (spectral) geometry of hyperbolic surfaces and how linear programming methods based on the Selberg trace formula can help. I will not assume any familiarity with hyperbolic geometry or the Selberg trace formula. This is joint work with Maxime Fortier Bourque.


Peter Sarnak  Princeton University and IAS Princeton  *Online* Prescribing the spectrum of locally homogeneous geometries 
[Abstract]
We review some recent developments (conformal bootstrap and random covers) concerning the bassnote spectrum of invariant operators on locally homogeneous geometries and in particular hyperbolic manifolds and of large 3regular graphs. Of special interest are rigidity features for creating spectral gaps.


Julia Slipantschuk  University of Warwick  Resonances for Anosov diffeomorphisms on the torus 
[Abstract]
I will present a complete description of PollicottRuelle resonances for a class of rational Anosov diffeomorphisms on the twotorus. This allows us to show that every homotopy class of twodimensional Anosov diffeomorphisms contains (nonlinear) maps with the sequence of resonances decaying stretchedexponentially, exponentially or having only trivial resonances.


Nikhil Srivastava  UC Berkeley  Many Nodal Domains in Random Regular Graphs 
[Abstract]
A nodal domain of a Laplacian eigenvector of a graph is a maximal connected component where it does not change sign. Sparse random regular graphs have been proposed as discrete toy models of "quantum chaos", and it has accordingly been conjectured by Y. Elon and experimentally observed by Dekel, Lee, and Linial that the high energy eigenvectors of such graphs have many nodal domains.


Yunhui Wu  Tsinghua University  *Online* Recent developments on random hyperbolic surfaces of large genus 
[Abstract]
In this talk, we report several very recent asymptotic results on certain classical geometric quantities viewed as random variables on the moduli space of Riemann surfaces for large genus (and many cusps). This talk is based on several joint works with Hugo Parlier, Xin Nie, Yang Shen and Yuhao Xue.


Nina Zubrilina  Princeton University  Root Number Correlation Bias of Fourier Coefficients of Modular Forms 
[Abstract]
Limiting distributions of spectral parameters of modular forms, in the prime, weight, and level aspects, are all well known to have mean 0. Nonetheless, it was recently discovered by Sutherland that when one restricts to forms of a prescribed root number, there are gaps between the means of such averages and 0. These computations extend a recent machine learning based study of He, Lee, Oliver, and Pozdnyakov, who observed a striking
oscillating pattern in the average value of the Pth Frobenius trace of elliptic curves of
prescribed rank and conductor in an interval range. In my talk, I will point to a source of this phenomenon in the case of all forms and compute the correlation function exactly.
