In the first part of the course, we will start with an introduction to the Gaussian free field (GFF), which is an object which has been at the heart of some recent groundbreaking developments in probability theory and its connections to many other branches of mathematics (e.g. complex analysis, geometry, partial differential equations) and statistical physics. Loosely speaking, GFF is a collection of Gaussian random “variables“ indexed by a two-dimensional domain with a prescribed covariance structure.
We will then discuss a few applications to the emerging theory of Liouville quantum gravity (LQG), which provides the most natural way of choosing a two-dimensional random surface (just like, if we are given a finite set X, the easiest way to choose a random element of X is uniformly, i.e. by assigning equal probability to each element of X). Mathematically, LQG is related to the problem of giving a meaning to the "exponential of the Gaussian free field". We will then see how LQG appears as the scaling limit of random planar maps (just like Brownian motion appears as the scaling limit of simple random walks, and many other discrete models).
The course will be kept at a very basic level and would require only some background from measure-theoretic probability. Further necessary material (e.g. tools from complex analysis and advanced probability) will be discussed in the lecture.
- Lehrende/r: Chiranjib Mukherjee
- Lehrende/r: Konstantin Julian Recke