Talks (titles and abstracts)

Yves Benoist: Convolution and square in cyclic group
Can one describe functions on a cyclic group of odd order whose convolution square f*f(2x) is proportional to the square f(x)2? We will see that this elementary question is related to abelian varieties with complex multiplication and their theta functions.

Martin Bridson: Profinite rigidity, central extensions, and finiteness properties
A finitely generated, residually finite group is said to be profinitely rigid if its set of finite quotients distinguishes it from all other finitely generated, residually finite groups. A few years ago, McReynolds, Reid, Spitler and I proved that certain arithmetic lattices in \({\rm{PSL}}(2,\mathbb{R})\) and \({\rm{PSL}}(2,\mathbb{C})\) are profinitely rigid.
In this talk, I shall sketch the proof of this and then explain how to extend the ideas to obtain infinite families of Seifert fibre spaces with profinitely rigid fundamental groups. For certain of these Seifert fibre spaces, the fundamental group \(G\) has the property that no other finitely presented, residually finite group has the same finite quotients as \(G\times G\), but there are infinitely many non-isomorphic finitely generated subgroups \(H<G\times G\) that have the same finite quotients as \(G\times G\). This is joint work with Alan Reid and Ryan Spitler.

Claire Burrin: Equidistribution, arithmeticity, and the (non)multiplicativity of Fourier coefficients of modular forms
I will discuss a qualitative proof, after Venkatesh and Ratner, of the failure of multiplicativity for the Fourier coefficients of modular forms attached to nonarithmetic Fuchsian groups, as well as a quantitative proof of this phenomenon involving the critical exponent of certain geometrically finite but infinite covolume Fuchsian groups.

Richard Canary: The skinning map and a lost theorem of Thurston
The skinning map records aspects of the internal geometry of a hyperbolic 3-manifold. We investigate the behavior of the skinning map on deformation spaces of hyperbolic 3-manifolds with freely indecomposable fundamental group. This investigation allows us to recover a proof of the generalized bounded image theorem of Thurston which played a crucial role in his proof of his hyperbolization theorem for atoroidal Haken 3-manifolds. Thurston's original proof appears to be lost in the mists of time. (Joint work with Ken Bromberg and Yair Minsky)

Indira Chatterji: Horospherical random graphs
We will describe a model of random graphs that resembles horospheres in a hyperbolic graph

Alex Eskin: Ruminations on a theorem of Furstenberg
A deep result of Furstenberg from 1967 states that if \(\Gamma\) is a lattice in a semisimple Lie group \(G\), then there exists a measure on \(\Gamma\) with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.

Tobias Hartnick: Discrete approximate subgroups of Lie groups - an invitation
We will survey some of the developments of the last decade in the theory of discrete approximate subgroups of Lie groups. We will explain how the focus has shifted from the spectral theory of quasicrystals to a new theory that combines methods from ergodic theory, dynamics and group theory in a way that is closely reminiscent in style to some of the work of Marc Burger and indicate some of his influences on the development of this theory. Based on joint work with Michael Björklund.

Dominique Hulin: Harmonic quasi-isometric maps
In a previous work, we proved that a quasi-isometric map \(f:X\to Y\) between two pinched Hadamard manifolds is within bounded distance from a unique harmonic map. In this talk, we will show that this result extends to maps \(f:\Gamma /X\to Y\), where \(\Gamma\) is a convex cocompact discrete group of isometries of \(X\), and \(f\) is locally quasi-isometric at infinity.

Fanny Kassel: Sharpness of proper cocompact actions and applications
We prove the so-called Sharpness Conjecture: any properly discontinuous and cocompact action of a discrete group on a real reductive homogeneous space G/H satisfies a strong form of properness called sharpness. As an application, for G/H of real corank one, such actions are characterized in terms of Anosov representations, and in particular they are stable under small deformations. Sharpness also allows us to deduce the nonexistence of proper cocompact actions on certain homogeneous spaces such as SL(2n,R)/SL(2n-k,R) for k = 1 or 2. Joint work with N. Tholozan.

Waltraud Lederle: Boomerang subgroups and the Stuck-Zimmer theorem
We introduce the notion of boomerang subgroups of a discrete group. Those are subgroups satisfying a strong recurrence property, when we consider them as elements of the space of all subgroups with the conjugation action. We prove that every boomerang subgroup of SL(n,Z) is finite or of finite index. Thus we give a new proof of the Stuck-Zimmer rigidity theorem for SL(n,Z) avoiding almost all measure theory. This is joint work with Yair Glasner.

Alex Lubotzky: Uniform stability of lattices in high-rank semisimple groups
Lattices in high-rank semisimple groups enjoy a number of special properties like super-rigidity, quasi-isometric rigidity, first-order rigidity and more. In this talk we will add another one: uniform ( a.k.a. Ulam) stability. Namely, it will be shown that (most) such lattices D satisfy: every finite-dimensional unitary "almost-representation" of D (almost w.r.t. to a sub-multiplicative norm on the complex matrics) is a small deformation of a true unitary representation. This extends a result of Kazhdan (1982) for amenable groups and or Burger-Ozawa-Thom (2013) for SL(n,Z), n>2. The main technical tool is a new cohomology theory ("asymptotic cohomology") that is related to bounded cohomology in a similar way to the connection of the last one with ordinary cohomology. The vanishing of H^2 w.r.t. to a suitable module implies the above stability. The talk is based on ongoing work with L. Glebsky, N. Monod and B. Rangarajan.

Rafe Mazzeo: When character varieties come to a good end
We consider the work of Taubes which leads to a theory of limits of diverging sequences of flat SL(2,C) connections (and other gauge theoretic equations associated to noncompact groups) in low dimensions. The limiting objects, called Z_2 harmonic spinors, are expected to play a role in wall-crossing formulae, but remain mysterious. I will describe recent work which focuses on the construction of new examples, deformability and index theory, particularly when the branching set is stratified. This is joint work with Haydys and Takahashi.

Shahar Mozes: Locally testable codes
A locally testable code is an error correcting code that has a property-tester who when receiving a word reads \(q\) bits of it that are randomly chosen, and rejects the word with probability proportional to its distance from the code. The parameter \(q\) is called the locality of the tester.
In a joint work with Irit Dinur, Shai Evra, Ron Livne and Alex Lubotzky, we introduce and use a certain family of complexes associated with groups and which have good expansion properties to construct an infinite family of locally testable codes which have constant rate, constant distance and constant locality.

Hee Oh: On Burger-Roblin measures
I plan to tell the story of how the Burger-Roblin measures came to be and how their lives took shape.

Frédéric Paulin: Pair correlations of logarithms of real and complex lattice points
We will study the correlations of pairs of complex logarithms of Z-lattice points in the real or complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We will prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of weights given by the Euler functions has applications to the pair correlation of the lengths of common perpendicular geodesic arcs between cusp neighborhoods in arithmetic hyperbolic manifolds. We will also answer a question of Pollicott and Sharp on the pair correlations of the lengths of closed geodesics in negatively curved manifolds. This is joint work with Jouni Parkkonen.

Andrés Sambarino: Exploiting an infinitesimal duality between projective structures and complex-hyperbolic structures
The purpose of the talk is to explain an infinitesimal version, obtained in collaboration with M. Bridgeman, B. Pozzetti and A. Wienhard, of classical results on the rigidity of Hausdorff dimension of limit sets, initiated by Bowen for quasi-Fuchsian manifolds and extended by Bourdon on the CAT(-1) setting.

Peter Sarnak: Prescribing the spectra of locally uniform geometries
After reviewing recent developments (conformal bootstrap and random covers) concerning the Laplace sepctra of hyperbolic manifolds and of large regular graphs, we focus on rigidity features to creating spectral gaps.

Barbara Schapira: Equidistribution of closed geodesics
On a compact negatively curved manifold, it is well known since Bowen and Margulis that periodic orbits of the geodesic flow are equidistributed towards the measure of maximal entropy of the geodesic flow, the so-called Bowen-Margulis-Sullivan measure. It allows to deduce a counting result about the asymptotics of the number of closed orbits of length at most T, when T goes to infinity.
When the manifold is not compact but the geodesic flow still admits a measure of maximal entropy, this result remains true. I will explain this work (in common with S Tapie) with examples where it applies.

Viktor Schröder: Boundaries of negatively curved spaces
We investigate geometric structures on the boundaries at infinity of negatively curved spaces. We focus on the inverse problem: reconstruction of the interior out of the boundary. In particular we recall results of Sergei Buyalo and discuss recent progress of Kingshook Biswas.

Anne Thomas: Fixed points for group actions on affine buildings
We prove a local-to-global result for finitely generated groups acting on 2-dimensional affine buildings. Our proofs combine general CAT(0) space techniques with building-theoretic arguments. This is joint work with Harris Leung, Jeroen Schillewaert and Koen Struyve.