Colloquium

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.

Extreme Value Statistics: An overview and perspectives

Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of `uncorrelated' variables is well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In thistalk, I will give an overview and perspectives on this interdisciplinary and rapidly evolving area of research.

Uniform exponential growth for CAT(0) cube complexes

Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has uniform exponential growth or it is virtually abelian. The behavior, in this sense, of a group that acts by isometries on a higher dimensional CAT(0) cube complex is not known. In this talk, I will present some generalizations of their theorem. On the one hand we allow the action to be proper instead of free and on the other hand we assume our space has isolated flats. I will define exponential growth and also present the general strategy to obtain a result like that of Kar--Sageev. This is joint work with Kasia Jankiewicz and Thomas Ng.

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