北京雁栖湖应用数学研究院

The modern theory of integrable systems, both classical and quantum, is well-known for its ability to provide unexpected connections between very different branches of mathematics and mathematical physics, such as representation theory, algebraic geometry, theory of special functions, and the list goes on and on. Many novel ideas in this field are motivated by questions in modern quantum field theory and string theory.
The goal of the proposed workshop is to bring together world-class leading experts, on one hand, and talented young researchers and graduate students, on the other, to discuss some recent advances and ideas, with the emphasis on the use of algebro-geometric methods and techniques in the theory of integrable systems. Some of the key themes of the workshop are connections between cluster integrable systems and Painlevé equations, BPS/CFT correspondence, Fredholm determinants and tau-functions.
We expect the workshop to provide a platform for an intensive exchange of ideas, updates on cutting-edge developments in the field, and creation of new research themes and international collaborations. To foster the latter, we plan to dedicate a significant amount of time to discussion sessions and open problem sessions.
Over the past several decades, algebraic geometry and integrable systems have influenced each other in significant and mutually beneficial ways. On the one hand, techniques from algebraic geometry—such as the study of moduli spaces, spectral curves, and theta-functions—have provided deep structural insights into classical and quantum integrable models (e.g. the Krichever construction, Hitchin systems, the KP hierarchy and systems of interacting particles, spin chains). This includes construction of Lax pairs of integrable systems that can then be used to obtain exact algebro-geometric solutions of soliton equations, definition of Hitchin integrable systems arising from the moduli spaces of stable holomorphic bundles, powerful tools in studying Painlevé equations and Painlevé transcendents, and many others. Conversely, phenomena discovered in integrable systems, isomonodromic deformations, and Painlevé transcendents—have inspired new questions about the geometry of algebraic varieties, singularity theory, and mirror symmetry. Moreover, tau-functions of integrable hierarchies appear to be generating functions for many enumerative problems in algebraic geometry, such as Gromov–Witten invariants and Hurwitz numbers. The famous Novikov conjecture (Shiota theorem) characterizes the Jacobians of algebraic curves with the help of the KP hierarchy. Another important recent development is the application of algebraic geometry and integrable systems for a consistent formulation of a non-perturbative quantum field theories. One of the most important examples is the Seiberg–Witten theory for the low-energy effective actions and BPS states in the N=2 SUSY Yang-Mills theory in four macroscopic dimensions. It reformulates this close-to-realistic physical theory in terms of integrable systems and representation theory of infinite-dimensional algebras, while the non-perturbative transformations of the BPS spectra are governed by the cluster algebra relations.

Objectives:
This workshop will explore such interactions, highlighting both foundational results and emerging directions, such as
• cluster algebras and cluster integrable systems
• duality in integrable systems and algebraic geometry
• geometric theory of Painlevé equations
• isomonodromy transformations and tau functions
• BPS/CFT correspondence
• Lefschetz thimbles in QFT

This workshop will emphasize interactions between the world-leading experts and young researchers, including graduate students and postdocs, both from China and abroad. We expect that it would result in new and fruitful collaborations.