Colloquium

Translation surfaces and their connections

A translation surface is a Riemann surface equipped with a metric that is locally Euclidean away from finitely many points, which are cone points with angle an integer multiple of 2 pi. This talk will present connections that translations surfaces have with billiards in polygons, smooth flows on surfaces and geometric topology.

On Moduli Spaces of Cut-And-Project Quasicrystals II

Spectrum of the laplacian on self-similar groups and holomorphic dynamics

In this talk based on a joint work with Rostislav Grigorchuk and Mikhail Lyubich, I will explain how the calculation of the spectrum of the laplacian on the Grigorchuk group and the Hanoi group is related to the iteration of particular rational maps in two complex variables and their ergodic properties.

On Moduli Spaces of Cut-And-Project Quasicrystals I

A cut-and-project quasicrystal is a point set obtained from projecting a higher-dimensional lattice of R^(d+m) on a lower-dimensional space R^d.

I will talk about results obtained with Yotam Smilansky and Barak Weiss, in which we give a classification of measures on the space of quasicrystals invariant under the SLd(R) action.

For d>2 we also obtain Roger-type moment formulas and apply those to counting "patterns" arising in this non-aperiodic setting. I promise nice pictures and some details of the proofs.

Measurement phase transitions and the statistical mechanics of tree tensor networks

A many-body quantum system that is continually monitored by an external observer can be in two distinct dynamical phases, distinguished by whether or not repeated local measurements (throughout the bulk of the system) prevent the build-up of long-range quantum entanglement. I will describe the key features of such “measurement phase transitions” and briefly sketch theoretical approaches to their critical properties that make connections with topics in classical statistical mechanics, such as percolation and disordered magnetism. Finally I will discuss random tensor networks with a tree geometry. These arise in a simple limit of the measurement problem, and they show an entanglement transition that can be solved exactly by a mapping to a problem of traveling waves.

Rigidity of topological entropy for a family of generalized Bowen–Series maps

Given a closed, orientable surface of constant negative curvature, we study a family of generalized Bowen–Series boundary maps, with each map defined for a particular fundamental polygon for the surface and a particular multi-parameter. We prove the following rigidity result: the topological entropy is constant and depends only on the genus of the surface. This is in contrast to a previous result that measure-theoretic entropy varies greatly within Teichmüller space. We give explicit formulas for both of these entropies. The proofs of rigidity use conjugation to maps of constant slope. 

This work is joint with Svetlana Katok and Ilie Ugarcovici.

Metric Ergodicity and its applications in Rigidity Theory

 An action is called Metrically Ergodic (ME) if the acted space admits no equivariant maps to metric spaces. This is a far reaching strengthening of usual Ergodicity. For probability measure preserving actions, ME is equivalent to Weak Mixing, but in general it is different and could be thought of as a generalization of Mautner phenomenon. It is a remarkable fact that every locally compact second countable group admits an action which is both Amenable and Metrically Ergodic. For example, the celebrated fixed point theorem of Ryll-Nardzewski could be deduced easily from this fact. In another direction, it implies Super-Rigidity for Lattices in Products. It is also a main player in the proof of the Simplicity of the Lyapunov Spectra in various situations. In my talk I will provide a gentle introduction to ME and discuss its applications.

The talk is based on a joint work with Alex Furman.

Some variants of the Rogers integral formula

The Rogers integral formula from the 1950's, which provides statistical information on points of the random lattice, have recently become a popular device in number theory and dynamics. With the revived interest in the formula also came its numerous variants and extensions. In this talk, I would like to discuss a few such formulas by myself, one concerning the rational points of the Grassmannians, and another generalizing the Rogers formula over the adeles of number fields. One immediate application is the non-Poissonian behavior of certain families of random sets of points, unlike in the much-studied case of the random lattice points in R^n.

Farey Sequences for Thin Groups

The study of thin groups has become a hot topic at the centre of a great deal of modern research. Recently the measure theory on infinite hyperbolic groups has opened these groups up to the techniques of homogeneous dynamics. In this talk I will give a brief and incomplete survey of this development and then explain a recent result about the local statistics of orbits by these groups and the connection to local statistics of some sequences of numbers. If there is time I will explain the connections between these theorems, Diophantine approximation on fractals, Ford packings, and continued fractions.

Discrete stationary random subgroups and application to discrete subgroups of Lie groups

The notion of invariant random subgroups (IRS) has proven extremely useful during the last decade, particularly to the study of asymptotic invariants of lattices. However, the scope of problems that one can investigate when restricting to invariant measures (on the space of subgroups)is limited. It was recently realised that the notion of stationary random subgroups (SRS), which is much more general, is still extremely powerful and opens up new paths to attacking problems that previously seemed to be out of our reach. 

  In this talk, using the notion of SRS, I will explain a proof of the following conjecture of Margulis: Let G be a higher rank simple Lie group and Λ ⊂ G a discrete subgroup. Then the orbifold Λ\G/K has finite volume if and only if it has bounded injectivity radius. This is a far-reaching generalisation of the celebrated Normal Subgroup Theorem of Margulis, and while it is new even for subgroups of lattices, it is completely general.    

This is a joint work with Mikolaj Fraczyk.