Tag - Integrable systems

Travis Scrimshaw: An Overview of Kirillov-Reshtikhin Modules and Crystals

19th January, 2024

Kirillov-Reshetikhin (KR) modules are an important class of finite-dimensional representations associated to an affine Lie algebra and the associated Yangian and quantum group. KR modules are known to appear in many integrable systems and govern the dynamics. In this talk, we will give an overview of the role KR modules play in the category of finite-dimensional representations, R-matrices and the fusion construction, their (conjectural) crystal bases, and how they relate to Demazure modules. In particular, we will focus on how to construct their crystal bases combinatorially and the different types of character theories. As time permits, we will discuss some of the relations with (quantum) integrable systems.

Vered Rom-Kedar: On Some Impact-like Hamiltonian Systems

4th December, 2023

The dynamics associated with mechanical Hamiltonian flows with smooth potentials that include sharp fronts may be modelled, at the singular limit, by Hamiltonian impact systems: a class of generalized billiards by which the dynamics in the domain’s interior are governed by smooth potentials and at the domain’s boundaries by elastic reflections. Results on persisting vs non-persisting dynamics of such systems will be discussed. In some cases, called quasi-integrable, the limit systems have fascinating behaviour: their energy surfaces are foliated by 2-dimensional level sets. The motion on each of these level sets is conjugated to a directed motion on a translation surface. The genus of the iso-energy level sets varies - it is only piecewise constant along the foliation. The metric data of the corresponding translation surfaces and the direction of motion along them changes smoothly within each of the constant-genus families. Ergodic properties and quantum properties of classes of such systems are established.

Alfonso Sorrentino: On the Rigidity of Integrable Twist Maps of the 2-Dimensional Annulus

27th November, 2023

In the study of Hamiltonian systems, integrable dynamics play a crucial role. Integrability, however, appears to be a delicate property that is not expected to persist under generic small perturbations. Understanding the essence of this fragility presents a compelling task, which turns out to be relevant across different contexts. In this talk, I shall present some results aimed at shedding more light on this issue, within the framework of symplectic twist maps of the 2-dimensional annulus. Specifically, I shall investigate the persistence and the properties of invariant Lagrangian tori that are foliated by periodic points and discuss how their fragility underpins the rigidity of completely integrable twist maps.

Franz Pedit: Minimal Lagrangian surfaces of high genus in CP2

24th August, 2023

The study of properties of surfaces in space has historically been a fertile ground for advances in topology, analysis, geometry, Lie theory, and mathematical physics. The most important surface classes are those which arise form variational problems, for example, minimal surfaces which are critical points of the area functional. The Euler Lagrange equations are PDEs which serve as model cases for developments in geometric analysis. Often these equations exhibit large (sometimes infinite dimensional) symmetry groups which puts the theory into the realm of integrable systems, that is, PDEs which allow for an infinte hierarchy of conserved quantities. This theory has been studied extensively over the past 40 years and led to significant advances in the classification of (minimal, constant mean curvature, Willmore etc.) surfaces of genus one. The higher genus case has been more illusive and examples are usually constructed using non-linear perturbation theory and gluing techniques.

In this talk I will explain how one can use ideas from integrable systems to construct examples of high-genus minimal Lagrangian surfaces without recourse to hard analysis.

This approach is more explicit than PDE existence results and one is able to obtain more quantitative information about the constructed examples, for instance, asymptotic area/energy estimates. I will also give a brief overview of the historical developments and the significance of minimal Lagrangian surfaces in mathematical physics.

Anton Izosimov: Dimers, networks, and integrable systems

18th March, 2022

I will review two combinatorial constructions of integrable systems: Goncharov-Kenyon construction based on counting perfect matchings in bipartite graphs, and Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on counting paths in networks. After that I will outline my proof of equivalence of those constructions.

Susan Tolman: Beyond semitoric

21st January, 2022

A compact 4-dimensional completely integrable system f: M→ℝ2 is semitoric if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of generates a circle action. Semitoric systems have been extensively studied and have many nice properties: for example, the preimages f−1(x) are all connected. Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitation. For example, there are blowups of S2×S2 with Hamiltonian circle actions which cannot be extended to semitoric systems. We expand the class of semitoric systems by allowing certain degenerate singularities, which we call ephemeral singularities. We prove that the preimage f−1(x) is still connected for this larger class. We hope that this class will be large enough to include not only all compact 4-manifolds with Hamiltonian circle actions, but more generally all complexity one spaces.

Vladimir Bazhanov: Quantum geometry of 3-dimensional lattices

26th October, 2021

In this lecture I will explain a relationship between incidence theorems in elementary geometry and the theory of integrable systems, both classical and quantum. We will study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices, lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analogue of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.

Agata Smoktunowicz: Some questions related to nilpotent rings and braces

27th February, 2021

In around 2005, Wolfgang Rump introduced braces, a generalization of nilpotent rings to describe all involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. This formulation then rapidly found application in other research areas. This talk will review these applications.

Definition. A set A with binary operations of addition +, and multiplication ∘ is a brace if (A, +) is an abelian group, (A, ∘) is a group and a ∘ (b+c)+a = a ∘ b+a ∘ c for every a, b, c ∈ A. It follows from this definition that every nilpotent ring with the usual addition and with multiplication a ∘ b = ab + a + b is a brace.

Braces have been shown to be equivalent to several concepts in group theory such as groups with bijective 1-cocycles and regular subgroups of the holomorph of abelian groups. In algebraic number theory there is a correspondence between braces and Hopf-Galois extensions of abelian type first observed by David Bachiller. There is also connection between R-braces and pre-Lie algebras discovered by Wolfgang Rump in 2014. One generator braces have been shown to describe indecomposable, involutive solutions of the Yang-Baxter equation.

On the other hand, Anastasia Doikou and Robert Weston have recently discovered some fascinating connections between braces and quantum integrable systems. In particular, to find solutions of the set-theoretic reflection equation it is needed to solve problems on some polynomial identities in nilpotent rings. Because previously the theory of polynomial identities was mainly developed for prime rings, and for the reflection equation we only consider nilpotent rings, there are no known methods for solving such problems. We will mention some open problems on polynomial identities in nilpotent rings which appear in this situation.

Bernard Leclerc: Cluster algebras and quantum loop algebras

16th August, 2016

In 2012, Hernandez and Jimbo introduced a new tensor category of representations of a Borel subalgebra of a quantum loop algebra, and classified its simple objects. This category contains the finite-dimensional representations of the quantum loop algebra, together with some new infinite dimensional representations. The motivation of Hernandez and Jimbo came from mathematical physics, in particular from papers of Bazhanov et al. where some examples of these new representations were used to define analogues of Baxter’s Q-operators in conformal field theory. Recently, using this new category, Frenkel and Hernandez were able to prove a long-standing conjecture of Frenkel and Reshetikhin on the spectra of the transfer matrices of some quantum integrable systems associated with quantum loop algebras. In this talk, I will explain that the new category of Hernandez and Jimbo fits very well with cluster algebras. More precisely I will show that cluster structures occur naturally in its Grothendieck ring, and can be helpful in finding new interesting functional relations. This is a joint work with David Hernandez.