Titles and abstracts
Trainer: Mikhail Bershtein
Title: Cluster integrable systems
Abstract: Cluster integrable systems form a relatively new, interesting, and important class of integrable systems. One of their basic features is that they are multiplicative (in more physical words relativistic). Another important feature is the natural constructions of discrete flows and quantization. Perhaps the most important application is the (conjectural) structure of cluster integrable system on the Coulomb branches of 4d supersymmetric theories.
A preliminary plan of the course (aka list of keywords): open Toda system, the definition of cluster varieties, cells in Poisson-Lie group, closed Toda system, spectral curves, dimer models, quantization of cluster varieties, quantum Toda systems. The course will be based on the works of Fock, Goncharov, Marshakov, Kenyon, Schrader, Shapiro.
Trainer: Veronica Fantini
Title: Introduction to summability and resurgence for irregular singular ODEs
Abstract: Solving ODEs in complex domain with irregular singularities can be done either formally or analytically. At the formal level, a method by Poincaré allows to build a frame of formal (and divergent) solutions. Then, Borel-Laplace summation allows to reconstruct an analytic frame of solutions by "re-summing" the formal ones. Alternatively, we can build a frame of analytic solutions by turning the ODE into an integral Volterra type equation. In the first two lectures we will discuss both approaches for a class of ODEs called of level 1. Then, in the third lecture will discuss some basic aspects of the theory of resurgence introduced by Écalle. In particular, we will study the properties of resurgent series that arises from solutions of irregular singular ODEs.
Trainer: Davide Masoero
Title: A primer on the complex WKB method with applications to the ODE/IM correspondence
Abstract: The complex WKB method is a powerful instrument to deduce and then prove asymptotic expansions of solutions to second-order linear differential equations (ODEs) with an analytic potential, as well as asymptotic expansions of the associated (generalised) monodromy data, in a great variety of asymptotic regimes.
In these lectures, we will be interested in studying the asymptotics, in the large energy or large momentum regimes, of the spectral determinants of a class of anharmonic oscillators, known as monster potentials, dual to the Bethe States of Quantum KdV model (ODE/IM correspondence),
Using this class of potentials as our guiding example, we will develop the complex WKB theory from scratch.
Plan of the lectures:
1. Review: Basic concepts of asymptotic analysis and ODEs in the complex plane.
The fundamental theorem of the complex WKB method.
2. Local structure of (horizontal) trajectories of a quadratic differential. Construction of subdominant solutions via the complex WKB method.
3. Central and lateral connection problems for monster potentials. Asymptotics of the spectral determinants.
Prerequisites: Complex Analysis and ODEs (mandatory), Functional Analysis and Riemann Surfaces (recommended).
Trainer: Andrey Smirnov
Title: Geometric methods in theory of integrable spin chains
Abstract: In this lecture series I overview an approach to the integrable systems of spin chains using equivariant enumerative geometry of quiver varieties. In this approach the Hilbert space of a spin chain is realized as the equivariant cohomology or K-theory of a quiver variety and the Hamiltonians of the spin chain feature as operators of quantum multiplication in these rings.
I will start the lectures by reviewing the most classical sl(2) XXX spin chain. Then I will introduce quiver varieties of type A, overview their basic properties and their equivariant cohomology/K-theory rings. After this, I will talk about equivariant counts of rational curves in quiver varieties, associated quantum difference equations and explain connections with the spin chains and Bethe ansatz.