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Organizers

Rod Halburd, University College London

Yang Shi(施洋), Flinders University

Da-jun Zhang(张大军), Shanghai University

Abstract

The theory of discrete integrable systems has undergone a truly remarkable development in the past three decades. Various novel mathematical techniques and methods have been developed to understand these discrete systems and their mathematical structures, bringing together ideas stemming from several branches of mathematics and physics, which are usually distinct, now come together: complex analysis, algebraic geometry, representation theory, spectral/isomonodromy analysis, random matrix theory, exactly solvable models, theory of special functions, graph theory, and difference geometry, etc. The workshop will bring together experts who have worked on various aspects related to the theory of discrete models and integrability, and thus aim at not only reviewing the remarkable progress in the past decade but also sorting out new emerging directions in discrete integrable systems.

Description of the aim

Historically speaking, the early examples of discrete integrable systems in the modern literature appeared in seminal work by Ablowitz and Ladik, and independently Hirota, in the 1970s, but earlier roots can be traced back to Bäcklund, Moutard and others, in the context of the 19th century differential geometry of curves and surfaces, and to the theory of Padé approximants and orthogonal polynomials, notably by Jacobi and Frobenius. In the past three decades, the study of discrete integrable systems has undergone a truly remarkable development, which brings together ideas stemming from several branches of mathematics and physics, which are usually distinct, now come together: complex analysis, algebraic geometry, representation theory, spectral/isomonodromy analysis, random matrix theory, graph theory, difference geometry, etc. Our first TSIMF workshop on the topic “Discrete Integrable Systems” (DIS-2016) was held successfully in 2016. In the past 10 years, there are many remarkable progress in this area, such as asymptotic analysis and birational geometry of discrete Painlevé, discrete integrability as multidimensional consistency, Lagrangian multiform theory, difference geometry and integrability, elliptic discrete integrable equations, algebra-geometry method to exact solutions, connections with numerical algorithms and combinatorics, integrable maps and cluster integrable systems, etc. There are also many new emerging directions which should be concerned about for the next ten years.  

Objectives: The workshop hopes to bring together leading experts who have worked on various aspects related to the theory of discrete models and integrability, and thus aim at not only reviewing the remarkable progress in the past decade, but also sorting out new emerging directions as well as outstanding problems in discrete integrable systems. We highlight following topics for this workshop:

• Discrete Painlevé equations: asymptotic analysis and birational geometry;

• Difference geometry and integrability;

• Lagrangian multiform theory and pluri-Lagrangian systems

• Algebro-geometric methods of discrete integrable systems;

• Elliptic discrete integrable equations;

• Spectral theory of difference operators;

• Cluster integrable systems;

• Quantization of classical integrable systems;

• Solutions of integrable lattice equations.