Workshop

We introduce Gaussian multiplicative chaos (GMC) and discuss its basic properties. The talks are intended especially to people having more an analysis background. No a priori knowledge of GMC is required.

The talk will review several approaches to constructing a Brownian motion on such spaces. Not only these spaces are equipped with a Fréchet topology, they might be viewed as both Riemannian and sub-Riemannian infinite-dimensional manifolds.

This is an introductory lecture on the Loewner energy and its links to Schramm-Loewner evolutions and Weil-Petersson quasicircles, a class of Jordan curves arising from the Kahler geometry on the universal Teichmuller space. This class of curves is also of interest in geometric function theory and has many drastically different but equivalent definitions. These links show that the Loewner energy sits at the crossroad of random conformal geometry and analysis. After explaining the basic definitions and the main connections, I will focus on the related open questions.

I will talk about a joint work with Jason Miller where we establish results on all geodesics in the Brownian map, including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each k=1,…,9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7,8,9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.

This will be a tutorial talk on the construction of fractal measures with an emphasis on Minkowski content measures.

I will give an elementary introduction to the Gaussian free field and some of its main properties (including conformal invariance in two dimensions, domain Markov property, and the behaviour of its circle average process). No advanced knowledge of probability will be assumed.

I will give an introduction to critical exponents in Bernoulli percolation by analyzing the model on the hierarchical lattice, a toy model that has much more symmetry than the Euclidean lattice and permits exact computation of several interesting quantities. Along the way I will review some classical topics such as the sharpness of the phase transition.

In this talk I will review the probabilistic construction of the Liouville theory (by F. David, C. Guillarmou, A. Kupiainen, V. Vargas and myself) on Riemann surfaces. In the second part of my talk I will review some recent progress related to the conformal bootstrap for the Liouville theory.

Ever since the seminal work of Ahlfors and Beurling in the 1950's, the study of removable planar sets with respect to various classes of holomorphic functions has proven over the years to be of fundamental importance for a wide variety of problems in complex analysis and related areas. Questions revolving around necessary and sufficient geometric conditions for removability have held a prominent role in the development of valuable techniques and applications.

In recent years, attention has been drawn to the more modern notion of (quasi)conformal removability, in view of applications to an ever-growing variety of central problems in complex analysis, probability and dynamics. In this talk, I will discuss various results related to conformal removability, focusing on applications to conformal welding and to Koebe's uniformization conjecture.

For a graph G, the chromatic polynomial at negative integer values is the expected value of a certain quantity, related to Dirichlet energy, over the simplex of random positive conductances (of sum 1). Talk based on joint work with Wei Yeung Lam.