# Overview

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#### Integrable systems, exactly solvable models and algebras

September 19-october 7, 2022

This concentration month will mainly be focused on four topics: (1) Box and ball systems; (2) Multivariate polynomials and exactly solvable models; (3) Reflection algebras, the q-Onsager algebra, the Heun-Askey-Wilson algebra and integrable systems; (4) Adelic Grassmanian, τ function, enumerative problems.

Box and ball systems

The box-ball system (BBS) discovered by Takahashi-Satsuma is one of the most basic ultradiscrete integrable systems and can be discussed from various viewpoints such as crystal bases of quantum algebras, tropical geometry, combinatorics, and cellular-automaton. In particular, as a new perspective, discussions and analyzes from probability theory and combinatorics are being actively conducted. The viewpoints of Pitman’s transformation in probability theory for discrete integrable systems enable to consider the behavior of the random initial state of the BBS. A workshop on BBS will provide an opportunity for direct discussions among researchers pursuing related work and will result in a strong push for this new development.

Multivariate polynomials and exactly solvable models

Historically, a significant part of the theory of classical orthogonal polynomials was developed in connection with its relevance for the solution of spectral problems in one-dimensional Mathematical Physics. More recently, prominent bases for the algebra of symmetric polynomials have been identified as eigenstates of quantum integrable particle models on the line or on the circle. The plan is to bring together experts from areas characterized by a fruitful interplay between the modern theory of multivariate orthogonal polynomials and their applications related to the study of quantum integrable particle systems, algebraic and coordinate Bethe Ansatz models, integrable probability and random matrices, exactly solvable spin chains, the combinatorics of symmetric functions, and (non)symmetric Macdonald polynomials (amongst others).

Reflection algebras, the q-Onsager algebra, the Heun-Askey-Wilson algebra and integrable systems

In the context of quantum integrable systems, Sklyanin’s reflection algebras have played an important role in recent years. Since the 90’s K-matrix solutions of reflection algebras have been the basic building blocks for constructing transfer matrices and the associated quantum spin chain Hamiltonians with integrable boundary conditions. The Bethe equations and the underlying TQ relations of these open quantum systems have been studied extensively. However, the interest in reflection algebras extends beyond this application: besides the fact that a complete classification of universal K-matrices is an active field of research in mathematics, it is now clear that reflection algebras provide an efficient framework for studying the representation theory of the (Heun)-Askey-Wilson algebras, q-Onsager algebra and its higher rank analogs. Also, techniques such as the algebraic Bethe ansatz and its modified version can be used to characterize the spectral properties of various operators emerging from these algebras. This is closely related with the subject of Leonard pairs and tridiagonal pairs and associated polynomial schemes. In particular, it is expected that reflection algebras and K-matrices associated with higher rank finite Lie algebras will naturally lead to generalizations of orthogonal polynomials. Also, establishing the precise relationship between Bethe ansatz equations and the representation theory for these algebras is expected to provide new insight on multivariate orthogonal polynomials.

Adelic Grassmanian, τ function, enumerative problems

The development of the theory of integrable systems is deeply tied with the geometry of Grassman manifolds since very early work of Sato, Segal, Wilson. In this context the notion of Tau function appeared in origin as generating function of commutative flows. Tau functions can be considered as a far-reaching generalization of the Riemann Theta function in the sense that in several contexts their vanishing characterizes the obstruction to the solvability of an associate linear problem, deeply related to the notion of Lax pair. Since their introduction, the range of applications of Tau functions associated to various integrable systems (like Kadomtsev–Petviashvili, Korteweg-de Vries and generalizations thereof) has expanded well beyond the original purview. Applications have been found in Random Matrix Theory and the associated theory of multi-orthogonal polynomials, enumerative geometry, combinatorics, symplectic geometry, theory of isomonodromic deformations, integrable probability.

One week of the workshop will celebrate the work of John Harnad, whose inspiring activity in the area of integrable systems and applications of the theory of tau functions to several problems has spanned three decades.