授课人：Maxime Fairon

授课方式：Zoom会议

课程简介：

Given an associative algebra, we can naturally obtain a corresponding commutative algebra using the representation functor. In 2008, Van den Bergh introduced the notion of double Poisson algebra which induces a classical Poisson algebra under this functor. An analogous structure was defined by De Sole, Kac and Valeri in 2015 to induce a Poisson vertex algebra at the commutative level. This mini-course will present the basics for the theory of double Poisson (vertex) algebras, construction and reduction techniques, and how these new structures are related. We shall also cover applications of this formalism to study integrable Hamiltonian systems of ODEs (e.g. Calogero-Moser systems) and integrable hierarchies of PDEs. Some open problems will also be presented.

Tentative schedule:

Lecture 1. Double Poisson algebras : theory

Lecture 2. Integrable particle systems from double Poisson algebras

Lecture 3. Double Poisson vertex algebras : theory

Lecture 4. Integrable non-abelian PDEs from double Poisson vertex algebras

课程安排：

授课人简介：

Maxime Fairon is Maître de Conférences (equivalent to lecturer) at Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France. His research interests are split between: i) classical integrable systems appearing in mathematical physics, and ii) non-commutative Poisson geometry based on double brackets.