Colloquium

On geometrically finite degenerations

The Sullivan dictionary provides a conceptual framework for understanding the connections between dynamics of rational maps and Kleinian groups. In this talk, I will discuss some recent development of geometrically finite degenerations of rational maps motivated by the dictionary. 

In particular, I will explain how to use this theory to understand the bumping structure of hyperbolic components and how to construct self-bumps, a phenomenon first discovered by McMullen on the Bers boundary and later generalized and studied by Bromberg, Anderson, Canary and McCullough.

I will also talk about how the analogues of a double limit theorem and Thurston’s compactness theorem can be deduced for this setting.

Jackiw Teitelboim Gravity and Random Matrix Theory

Two dimensional gravity is a fascinating subject of interest in the study of quantum gravity, statistical mechanics and random matrix theory. In this talk we will introduce a model of two dimensional gravity called Jackiw-Teitelboim gravity which has received considerable attention recently, motivated partly by the study of spin systems in condensed matter physics. We will discuss the classical solutions of the theory and quantise it using the path integral formalism. It will turn out that the theory is equivalent to a quantum mechanical system with a random Hamiltonian. The connection will involve recursion relations which were obtained by Mirzakhani in her study  of the moduli space of bordered Riemann surfaces. 

      The talk is intended  to be non-technical and accessible to a broad audience. It will be based on the following three papers:

1) D. Stanford and E. Witten, ``Fermionic Localisation of the Schwarzian Theory", arXiv: 1703.04612 

2) P. Saad, S. Shenker and D. Stanford, ``JT gravity as a Matrix Integral", arXiv: 1903.1115 

3) U. Moitra, S. Sake and S. P. Trivedi, ``JT gravity in the second order formalism", arXiv: 210.00596   

A universal Cannon-Thurston map and the surviving complex

The fundamental group of a surface (closed or with punctures) acts on the curve complex of the surface with one additional puncture via the Birman Exact Sequence.  I will describe a construction of a continuous, equivariant map from a subset of the circle at infinity of the universal cover of the surface onto the Gromov boundary of the curve complex (along the way, explaining what these objects and actions are).  This map is universal with respect to all Cannon-Thurston maps coming from type-preserving Kleinian representations without accidental parabolics.  In the case of closed surfaces, this map was constructed in joint work with Mj and Schleimer, and in this talk I will talk about an extension to the case of punctured surfaces obtained in joint work with Gultepe and Pho-On.  The proof for punctured surfaces involves constructing a continuous equivariant map to the Gromov boundary of a "larger" complex called the surviving complex.  I will describe this complex, its Gromov boundary, and the construction of the map.

Gromov compactness revisited

Gromov's compactness theorem for pseudoholomorphic curves is a fundamental result in almost-complex geometry which finds many applications in symplectic topology. The usual proofs of this theorem show sequential compactness of the relevant moduli space. I will sketch the proof of a quantitative version of this theorem, blackboxing some of the analytical estimates and focusing on the more combinatorial aspects of the proof. At the end, I will also outline an application (in progress) of this quantitative compactness theorem.

Dynamics on the moduli space of point configurations on the Riemann sphere

A rational function f(z) is called post-critically finite (PCF) if every critical point is either pre-periodic or periodic. PCF rational functions have been studied for their special dynamics, and their special distribution within the moduli space of all rational maps. By works of W. Thurston and S. Koch, (“almost") every PCF map arises as an isolated fixed point of an algebraic dynamical system on the moduli space M_{0,n} of point-configurations on P^1; these dynamical systems are called Hurwitz correspondences. I will give an overview of the study of PCF rational maps, introduce Hurwitz correspondences and present results on their dynamics.

Rotational invariance in planar percolation

In this series of two talks, we will discuss the rotational invariance of critical Bernoulli percolation on the square lattice. We will start by reviewing the state of the art on criticality for this very classical model, and will then discuss the recent progress in the understanding of the large scale properties of the model at criticality.

Emergent symmetries in statistical physics systems

A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millenium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

The word and Riemannian metrics on lattices of semisimple groups II

The word and Riemannian metrics on lattices of semisimple groups I

Patterson-Sullivan measures for Anosov subgroups II