There will be two main courses lasting for the entire school, and three mini-courses, Lectures will be taped and posted on MathTube.
Ivan Corwin: Interacting particle systems, growth models, stochastic PDEs and directed polymers through the lens of the stochastic six vertex model.
Introduced by Gwa and Spohn in 1992, the stochastic six vertex model is a cousin of the symmetric six vertex model introduced much earlier by Pauling in 1935. Through limit transitions and various specialization of parameters, this models can be related to a rich hierarchy of probabilistic systems including the asymmetric simple exclusion process, Kardar-Parisi-Zhang (KPZ) stochastic PDE, stochastic telegraph equation, and certain special models of random walks in random environments, directed polymers and last passage percolation.
This course will focus on advances in our understanding of the above models, through the lens of the stochastic six vertex model. We will explain how old tools such as the Bethe ansatz, Yang-Baxter equation, fusion, symmetric functions and Markov dualities have gained new probabilistic applications and interpretations. Using these tools we will extract precise asymptotics of these systems related to both the Kardar-Parisi-Zhang universality class and stochastic PDE, and uncover other remarkable probabilistic properties.
Frank den Hollander: Metastability for interacting particle systems
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to the action of temporal random forces, typically referred to as noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition, i.e., a crossover that involves a jump in some intrinsic physical parameter, such as the energy density or the magnetisation. Attempts to understand and model metastable systems mathematically go back to the 1930's. The modern mathematical approach to metastability dates from around 1970.
One approach to metastability is via the theory of large deviations in path space. The realisation that metastable behaviour is controlled by large deviations of the random processes driving the dynamics has permeated most of the mathematical literature on the subject. The present mini-course focusses on an alternative way to tackle metastability, which initiated around 2000. It interprets the metastability phenomenon as a sequence of visits of the path to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets with the help of potential theory. The key point is the realisation that, in the specific setting related to metastability, most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially be understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle.
The setting of this mini-course is the theory of reversible Markov processes. Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. The models we aim at range from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour. Along the way we will encounter a variety of ideas and techniques from probability theory, analysis and combinatorics, including martingale theory, variational calculus and isoperimetric inequalities. It is the combination of physical insight and mathematical tools that allows for making progress, in the best of the tradition of mathematical physics.