Course Materials

The three lecture courses will be:

Nathanaël Berestycki: Two-dimensional Liouville quantum gravity and the Gaussian free field. I will describe recent developments connected with two-dimensional Liouville quantum gravity.  After giving some background detailing the general problem and the motivations coming from the critical behaviour of 2d statistical mechanics models, I will outline two different approaches that have emerged recently to tackle the problem from a rigorous standpoint.  The first is based on a discretisation of space via random planar maps and the other tries to seek directly a continuous formulation based on the Gaussian Free Field; I will focus mainly on the latter.  Topics to be discussed include in particular a gentle introduction to the GFF and that associated Liouville measure, Kahane's Gaussian multiplicative chaos theory, the KPZ formula, the quantum gravity zipper of Scott Sheffield, and Liouville Brownian motion.

Paul Bourgade: Random matrices and PDEs.  Eugene Wigner has envisioned that the distributions of the eigenvalues of large Gaussian random matrices are new paradigms for universal statistics of large correlated quantum systems.  These random matrix eigenvalues statistics supposedly occur together with delocalized eigenstates.  In this course, I will explain recent developments proving this paradigm for many matrix models.  The emphasis will be on the interplay between a local quantum unique ergodicity (complete delocalization of eigenvectors) and universality (repulsion between eigenvalues).  Important tools will be random walks in dynamic random environments, Hölder regularity and homogenization theory for PDEs.

Ivan Corwin: Integrable probability and the KPZ universality class.  I will discuss recent developments in the theory of integrable probability, in particular that which relates to the Kardar-Parisi-Zhang universality class.  An emphasis will be placed on understanding how integrable structures translate into new and interesting probability.  The two main structures which will be leveraged are symmetric function theory (e.g. Macdonald and related polynomials) and quantum integrable systems (e.g. coordinate and algebraic Bethe ansatz).  Probabilistic systems discussed with include particle systems, growth processes, directed polymers, stochastic PDEs, random matrices and tilings.


PREPARATORY MATERIALS:

Ivan's lectures will be based on these notes (http://www.math.columbia.edu/~corwin/MSRIJuly2014.pdf) which accompanied a longer summer school course he gave last summer at MSRI (videos and other background info online available at http://www.msri.org/summer_schools/714).

Paul would like to suggest Laszlo Erdos' ICM proceedings (http://arxiv.org/pdf/1407.5752v2.pdf) as preparatory reading.

Nathanaël's course notes can be found at the following link http://www.statslab.cam.ac.uk/~beresty/Articles/oxford.pdf