Colloquium

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).

Convex Hulls of Two Dimensional Stochastic Processes

Convex hull of a set of points in two dimensions roughly describes the shape of the set. In this talk, I will discuss the statistical properties of the convex hull of several stochastic processes in two dimensions. By adapting Cauchy's formula to random curves, we develop a formalism to compute explicitly the mean perimeter and the mean area of the convex hull of arbitrary two dimensional stochastic processes of a fixed duration. Our result makes an interesting and general connection between random geometry and extreme value statistics. I will discuss two examples in detail (i) a set of n independent planar Brownian paths (ii) planar branching Brownian motion with death. The first problem has application in estimating the home range of an animal population of size n, while the second is useful to estimate the spatial extent of the outbreak of animal epidemics. Finally I will also discuss two other recent examples of planar stochastic processes: (a) active run-and-tumble process and (b) resetting Brownian motion.

Yang-Mills on the lattice: New results and open problems

Quantum Yang-Mills theories have mathematically well-defined formulations on lattices, known as lattice gauge theories. I will give a brief introduction to lattice gauge theories and a survey of existing results, followed by an overview of a number of longstanding open problems and recent progress on some of these questions.

CAT(0) cube complexes for the working geometer

Four-point functions in the Fortuin-Kasteley cluster model

The determination of four-point correlation functions of two-dimensional lattice models is of fundamental importance in statistical physics. In the limit of an infinite lattice, this question can be formulated in terms of conformal field theory (CFT). For the so-called minimal models the problem was solved more than 30 years ago, by using that the existence of singular states implies that the correlation functions must satisfy certain differential equations. This settles the issue for models defined in terms of local degrees of freedom, such as the Ising and 3-state Potts models. However, for geometrical observables in the Fortuin-Kasteleyn cluster formulation of the Q-state Potts model, for generic values of Q, there is in general no locality and no singular states, and so the question remains open. As a warm-up to solving this problem, we discuss which states propagate in the s-channel of such correlation functions, when the four points are brought together two by two. To this end we combine CFT methods with algebraic and numerical approaches to the lattice model. We then outline work in progress that aims at solving the problem entirely, through an interchiral conformal bootstrap setup that makes contact with time-like Liouville field theory and a number of profound algebraic results.

Logarithmic correlations in percolation and other geometrical critical phenomena

The purpose of renormalisation group and quantum field theory approaches to critical phenomena is to diagonalise the dilatation operator. Its eigenvalues are the critical exponents that determine the power law decay of correlation functions. However, in many realistic situations the dilatation operator is, in fact, not diagonalisable. Examples include geometrical critical phenomena, such as percolation, in which the correlation functions describe fluctuating random interfaces. These situations are described instead by logarithmic (conformal) field theories, in which the power-law behavior of correlation functions is modified by logarithms. Such theories can be obtained as limits of ordinary quantum field theories, and the logarithms originate from a resonance phenomenon between two or more operators whose critical exponents collide in the limit. We illustrate this phenomenon on the geometrical Q-state Potts model (Fortuin-Kasteleyn random cluster model), where logarithmic correlation functions arise in any dimension. The amplitudes of the logarithmic terms are universal and can be computed exactly in two dimensions, in fine agreement with numerical checks. In passing we provide a combinatorial classification of bulk operators in the Potts model in any dimension.

Bose Fermi duality in matter Chern Simons theories

In four and higher spacetime dimensions Bosons and Fermions are irreducibly different. This difference blurs out in three spacetime dimensions. This blurring allows a nontrivial phenomenon. There is now substantial evidence that Bosons coupled to Chern Simons theories are dual to Fermions coupled to (level rank dual) Chern Simons theories. I will review what is understood - and what is not - about these dualities.