Colloquium

Invariant random subgroups and applications to lattices

In mathematics in general, it is fruitful to allow randomness. Indeed, it is often easier to deal with random rather than deterministic objects. It seems miraculous, however, when we are able to say more about deterministic objects by considering them as random ones.

The notion of invariant random subgroups (IRS) has proven extremely useful  to the study of asymptotic invariants of lattices. In this talk I will demonstrate several results (old and new) about lattices in semisimple Lie groups which are proved by considering them as IRS.

Estimating the Hausdorff dimension of Julia sets of holomorphic correspondences

We consider a special class of holomorphic correspondences which generalizes the quadratic family and use some tools of thermodynamic formalism to estimate the Hausdorff dimension of the Julia sets of this family of holomorphic correspondences.

Uniformization of quasiconformal trees

Quasisymmetric maps are generalizations of conformal maps and may be viewed as a global versions of quasiconformal maps. Originally, they were introduced in the context of geometric function theory, but appear now in geometric group theory and analysis on metric spaces among others. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetric to some model space. Here we consider this uniformization problem for certain trees. In particular, we consider the "continuum self-similar tree'' (CSST) and give necessary and sufficient conditions for another tree to be quasisymmetrically equivalent to the CSST. One motivation to study the CSST is that it is almost surely homeomorphic to the continuum random tree introduced by Aldous. This is joint work with Mario Bonk.

Subshifts of finite type with a hole: properties and applications

Dynamical systems can be broadly classified into closed and open systems. In a (traditional) closed system, the orbit of a point lies in the state space for all time, whereas in an open system, the orbit of a point may eventually escape from the state space through a hole. The notion of open dynamical systems was introduced by Pianigiani and Yorke in 1979, motivated by the dynamics of a ball on a billiard table with pockets. In this talk, we consider an irreducible subshift of finite type and study the average rate at which the orbits escape into the hole. This problem turns out to be an interesting application of a combinatorial problem which is counting the number of words of fixed length not containing any of the words from a fixed collection as subwords. We also present some applications of our results in computing the Perron eigenvalues and eigenvectors of any irreducible (0-1) matrix and obtaining a combinatorial expression for the Parry measure of a subshift of finite type.

Volume vs. complexity of hyperbolic groups

In this talk we will discuss the relation between the volume of a quotient X/G of a (Gromov) hyperbolic graph X by a group G acting freely and cocompactly on X, and the "complexity" of the group G. We will then show how to use this relation to study finite-index subgroups of cubulated hyperbolic groups.

Brolin’s theorem for finitely generated polynomial semigroups

In this talk, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. The existence of this measure follows from a very general result of Dinh--Sibony applied to a holomorphic correspondence that one can naturally associate with a semigroup of the above type. We are interested in a precise description of this invariant measure. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Our result generalizes the classical result by Brolin. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. If time permits, we shall also present some bounds on the capacity and diameter of the Julia sets of such semigroups, which uses the F-functional of Mhaskar and Saff.

Perturbing the action of a hyperbolic group on its boundary

A hyperbolic group G comes with an action by homeomorphisms on its Gromov boundary.  In general this boundary is some compact metrizable space which can have complicated local topology, but sometimes it is a manifold (for example if G is the fundamental group of a closed negatively curved manifold).  We show that the action of a torsion free hyperbolic group on its boundary is topologically stable, assuming that boundary is a manifold.  

This is joint work with Kathryn Mann.

Random trees from conformal welding

Conformal welding is a way of gluing Riemann surfaces along their boundary via a specified equivalence relation. Even in the case that the resulting boundary interface is a simple curve, the existence and uniqueness of the resulting conformal structure is in general difficult to determine; this is the conformal welding problem.

We give criteria for the solution of the welding problem in the case that the boundary interface is a dendrite. In particular we prove that a natural conformal welding problem associated with the continuum random tree (CRT) has a solution, giving rise to a `canonical’ embedding of the CRT in the plane.

Joint work with Steffen Rohde.

Hot spots problem for convex planar domains

In this talk I will first give a brief introduction to the hot spots problem for planar domains. I will then recall the old results on this problem, and, in the last part of the talk I will discuss some recent developments.

`Quantization' and Topological Aspects of the Space of Renormalization Group flows in 2-dim Quantum Field Theory

In this talk we will discuss a `quantization' of the renormalization group equations by adding a gaussian noise term and converting them into stochastic differential equations. We will discuss the case of two dim. unitary QFTs where a Zamolodchikov c-function exists and the `drift term' is a gradient of the c-function. Quantization leads to supersymmetric quantum mechanics which can be studied in the `semi-classical' approximation. In particular one can attempt to characterise the topology of the space of paths in the path integral  using Morse theory using the Zamolodchikov c-function as a Morse function. Assuming the validity of Morse inequalities in the infinite dimensional case we calculate, as an illustration, the Betti numbers of the space of flows of the c < 1 unitary minimal models of 2-dm conformal field theory. This talk is based on work with S. R. Das and G. Mandal: "Stochastic differential equations on 2-dim. theory space and Morse theory", Mod. Phys. Letts A, Vol 4 No.8 (1989).