ICTS

Integrable systems share the properties of being exactly solvable in some sense and of having many conserved quantities. Investigating their behavior is key to understanding the wealth of non-integrable models falling in the same universality class. While the first examples of integrable systems were continuous, a large array of discrete integrable systems have been discovered over the last 60 years. These discrete systems hail from various branches of theoretical physics (statistical physics, string theory) and mathematics (combinatorics, representation theory, geometry, probability). They all possess remarkable algebraic structures.

This program proposes to explore several interrelated aspects of discrete integrable systems. We will focus on three aspects that are currently active topics of research:

1. Integrable difference equations, their soliton solutions and the rich structure of their singularities. Ultradiscretization of these equations, yielding cellular automata (e.g. box-ball systems) which have recently been related to quantum groups, combinatorics, tropical geometry and stochastic processes.

2. The cluster algebra structure underlying many discrete integrable systems. Some of them related to geometric objects (Grassmannian, dynamics on polygons, discrete differential geometry) and some of them coming from mathematical physics (Y-systems, dimer model, Toda lattice).

3. Integrable probability: interacting particle systems, stochastic growth models related to the KPZ equation, random tilings, etc. Tools from quantum integrable systems (vertex models) and from the theory of symmetric functions.

The first week comprises three mini-courses with accompanying problem sessions. Junior participants will be given the possibility of presenting their work. The second week consists of a workshop.

Organized with support from the Japanese Society for the Promotion of Science (JSPS), the Centre National de la Recherche Scientifique and CEA Saclay (France).

Eligibility criteria: Researchers (junior and senior) interested in discrete integrable systems and related areas. Accommodation will be provided for outstation participants at our on-campus guest house.

ICTS is committed to building an inclusive environment, non-discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.