Integrable Many-Body Systems through the Mathematical Lens

ended

The project focuses on the relation between integrable many-body systems and three important fields of mathematics: partial differential equations, Painlevé equations, and probability theory. The goal is to build new connections in this context by considering recent results as follows. The first objective consists in explaining how a class of integrable many-body systems that appeared over the last 5 years can be used to parametrise specific solutions to hierarchies of partial differential equations. An algebraic and a geometric interpretation of these parametrisations will be sought. The second objective deals with the extension of the Hamiltonian formulation of many-particle Painlevé equations, called the Calogero-Painlevé correspondence, to new cases. This investigation will make a central use of the current activity on discrete Painlevé equations and the quantum analogues of Painlevé equations. The third objective related to probability theory is two-fold. On the one hand, quantum versions of integrable many-body systems will be derived by adding noise to specific diffusion processes that are constructed using their classical versions. On the other hand, important properties of probability distributions appearing in random matrix theory or the study of beta ensembles will be obtained. The point of view will be to interpret these distributions in terms of suitable many-body systems whose integrability will play a key role for the computation of the desired properties.