AGQ integrability seminar

This is an informal gathering of people interested in integrable systems from the University of Edinburgh, Heriot-Watt University and the University of Glasgow.

The seminar runs bi-weekly on Thursdays alternating between Edinburgh (Bayes Centre, room 5.46) and Glasgow (University of Glasgow, School of Mathematics and Statistics, usually room 311B).

Organizers: Sasha Shapiro (UoE), Yegor Zenkevich (UoE), Robert Weston (HWU), Alexander Cooper (HWU), Anton Izosimov (UoG), Jules Lamers (UoG), Christian Korff (UoG).

 

Next seminars:

May 14, 2026, University of Glasgow, School of Mathematics and Statistics, room 311B.

13:00-16:00 Lukas Hardi (University of Hamburg)
 

Spin Ruijsenaars-Schneider models are Coulomb branches

To every 4d N=2 quiver theory, one can associate its moduli space of Coulomb vacua, also called the K-theoretic Coulomb branch of the quiver, which carries the structure of a Poisson variety. In this talk, we show that the (abelianized) K-theoretic Coulomb branch of the necklace quiver is the complex integrable system known as the trigonometric spin Ruijsenaars–Schneider model. This is achieved by exhibiting a collection of L-operators associated to the arrows in the quiver, thereby defining an integrable lattice model. Time permitting, we discuss the quantization of the model.

 

Past seminars:

April 30, 2026, University of Glasgow, School of Mathematics and Statistics, room 110

13:00-16:00 Yegor Zenkevich (University of Edinburgh)

K-theoretic Coulomb branches, integrable systems and coproducts on shifted quantum affine sl(2)

Defining a quantum group involves choosing a coproduct. Different choices are related by Lusztig’s braid group action and at the same time by Drinfeld twists.
We study how twists and automorphisms act on the space of coproducts on the simplest (shifted) quantum affine algebra U_q(^sl(2)). We show that in a particularly simple representation given in terms of a quantum torus the corresponding Drinfeld twists diagonalise Hamiltonians of an open Toda integrable system providing an explicit pq-duality transformation. We also describe the relations between our construction, K-theoretic Coulomb branches and their cluster algebra structure.

 

March 19, 2026, University of Edinburgh, Lister Learning and Teaching Centre, 5 Roxburgh Pl, Edinburgh EH8 9SU, room LLC G.13.

13:00-15:00 Alexander Cooper (Heriot-Watt University)
 
TQ Relations Associated with the 6-Vertex Model
 
In the first part of the talk, I will remind the audience of some background regarding the (6-vertex) transfer matrix of the 1D quantum XXZ Heisenberg spin chain, and its connection to spin-1/2 representations of the quantum affine algebra Uq(^sl2).
 

I will then move on to discussing the diagonalization of this transfer matrix via Baxter’s auxiliary matrix approach, and the modern construction of the Q-operator which facilitates it, using infinite-dimensional q-oscillator representation(s) of the Borel subalgebras of Uq(^sl2).  I will discuss the derivation of the operatorial TQ relation via two different approaches, one which may naturally be viewed as a ‘limit’ of a fusion/bootstrap procedure, and one which is best viewed as a factorisation of a transfer matrix associated with an infinite dimensional representation of Uq(^sl2).

 

Should time further permit me, I will finally briefly outline the route from here to the analogous relations for the XXX chain, a route which equally applies to recent work involving similar constructions for spin chains with open (rather than (quasi-)periodic) boundary conditions.

 

 

February 5, 2026, University of Glasgow, room 133 of the Hetherington Building

13:00-15:45 Bart Vlaar (BIMSA)
 
On quantum groups, old and new
 
Quantum groups (= quantized universal enveloping algebras) were defined by Drinfeld and by Jimbo in the mid 1980s, generalizing a construction by Kulish and Reshetikhin. In addition to their origin in mathematical physics, they are objects of interest in (non-commutative) geometry, (harmonic and functional) analysis and (low-dimensional) topology, as well as pure representation theory. We will discuss a more recent, more general, but related family of algebras called quantum symmetric pairs or i-quantum groups, with a focus on the topological viewpoint and the application to quantum integrable systems. We will discuss some recent progress and some open problems. Mainly based on joint works with Andrea Appel, and with Alec Cooper and Robert Weston.

 

January 22, 2025, Edinburgh, Bayes Centre, ICMS Lecture Theatre 5.10

13:00-15:45 Christian Korff (University of Glasgow)
 
From q-bosons to q-Whittaker functions
 
The plan is to talk about some old work and unfinished business: the realisation of q-Whittaker polynomials (specialisations of Macdonald polynomials) as partition functions of an exactly solvable lattice model [defined in CMP Volume 318 (2013), Page 173-246]. The corresponding transfer matrix of the lattice model can be identified as a quantisation of the B”acklund transform for the Ablowitz-Ladik chain, a spatial discretisation of the nonlinear Schr”odinger model [JPA (2016) 49 104001].  In the q=0 limit one obtains for periodic boundary conditions a combinatorial model for the Verlinde algebra of the su(n)_k WZW CFT [C.K. and C. Stroppel, AIM 225, Issue 1, 10 September 2010, Pages 200-268].
 

December 4, 2025, Edinburgh (UNUSUAL LOCATION: G.02, 19 George Square)

13:00 Sasha Shapiro (University of Edinburgh)

Integrable systems, character varieties, and clusters

Phase spaces of many famous integrable systems, such as Toda, Ruijsenaars, Gelfand-Tsetlin, etc, can be realised as character varieties. In turn, cluster structure on the latter allows for combinatorial description of these integrable systems, and is helful in the study of their discrete dynamics and Hamiltonian eigenfunctions. During the talk I will discuss some of these topics focusing on low-dimensional examples.

 

November 6, 2025, Edinburgh (UNUSUAL LOCATION: room LT1, Appleton Tower)

13:00-13:45 Alec Cooper (Heriot-Watt University)
 
Preseminar: A Brief Discussion of Quantum Groups and Diagrams
 
14:15-15:45 Robert Weston (Heriot-Watt University)
 
The algebraic description of open quantum spin chains
 

October 23, 2025, Glasgow (UNUSUAL LOCATION: room 513, Boyd Orr building)

Misha Feigin (University of Glasgow)
 
Cherednik algebras and Calogero-Moser type systems
 
This is going to be an introductory/overview talk. I’m going to define rational Cherednik algebras, discuss their basic properties and representations, some important subalgebras, and related integrable systems.
 

October 9, 2025, Edinburgh (UNUSUAL LOCATION: Lecture Theatre A, 40 George Square)

13:00-13:45 Mikhail Vasiliev (University of Glasgow)

Introduction to cluster algebras
 
I will give a brief introduction to cluster algebras. We will discuss several simple examples to illustrate how the machinery of cluster algebras work. I will also introduce Poisson structure arising in cluster algebras, which will be used by Anton in his talk.
 

14:15-15:45 Anton Izosimov (University of Glasgow)

Cluster integrable systems 
 
We’ll start with discussing the notion of a cluster integrable system due to Goncharov and Kenyon, along with a few examples of such systems. Then I’ll explain why such systems can be viewed as “type A” objects and discuss how to construct their counterparts for (some) other Dynkin types.
 

September 25, 2025, Glasgow

Yegor Zenkevich (University of Edinburgh)

Introduction to quantum toroidal algebras and integrability

 

Seminar notes: