Schedule of the Workshop: Integrability - modern variations

Yaroslav Bazaikin: Globally hyperbolic Lorentzian spaces with special holonomy
The list of possible Lorentzian holonomy groups is known and all they are realized by locally defined "noncomplete" Lorentzian metric. We describe the globally hyperbolic Lorentzian manifolds with holonomy group from the part of this list.

Yuri Berest: Dixmier groups and the Dixmier conjecture
I will discuss a class of infinite-dimensional algebraic groups closely related to the group of polynomial automorphisms of the affine plane. Originating in the theory of integrable systems these groups have several different incarnations that allow one to describe their structure explicitly and even resolve the isomorphism problem. As a motivation I will explain a relation to the Jacobian Conjecture and its `quantum' counterpart - the Dixmier Conjecture.

Alexey Bolsinov: Holonomy groups in pseudo-Riemannian geometry and Euler--Manakov tops
A remarkable relationship has been recently observed between integrable systems on so(n) known as Euler-Manakov tops and curvature tensors of projectively equivalent Riemannian metrics. This relationship is used to construct a new class of holonomy groups in pseudo-Riemannian geometry.

Boris Dubrovin: Frobenius manifolds and integrable systems of topological type

Oliver Fabert: From Floer homology to integrable systems via SFT
Symplectic field theory (SFT) naturally assigns to each symplectic mapping torus (of a symplectomorphism on a closed symplectic manifold) an infinite number of commuting Hamiltonian systems. In the case when the underlying symplectomorphism is the identity, we know that this system of commuting Hamiltonians is complete, because it agrees with the integrable hierarchy for the underlying symplectic manifold from Gromov-Witten theory. In my talk I will present results towards proving or disproving completeness of the SFT Hamiltonian systems for general symplectic mapping tori, making use of the relation to the Floer theory of the underlying symplectomorphism. Since the latter generalizes the relation between SFT and Gromov-Witten theory in the case of the identity map, the idea is to find a natural generalization of the classical approach to integrable systems using Frobenius manifolds and bihamiltonian structures starting from the Floer theory of a symplectomorphism.

Giovanni Felder: Critical loci and the Batalin-Vilkovisky master equation
Batalin and Vilkovisky introduced a method to evaluate oscillatory integrals occuring in quantum field theory in the presence of symmetries of a very general kind. I will review this method and present a mathematical treatment in the finite dimensional case: starting from a polynomial in n variables (the classical action) one canonically produces a stable equivalence class of solutions of the master (or Maurer-Cartan) equation on an odd symplectic manifold and a corresponding cohomology, which comes with a cup product and a Poisson bracket of degree 1. I will discuss the meaning of this cohomology in low degree and ways to compute it, and present simple examples.
The talk is based on joint work with David Kazhdan.

Iain Gordon: Dunkl operators, Hecke algebras and Calogero-Moser spaces
I will explain the use of Dunkl operators, and more generally Cherednik algebras and generalized Calogero-Moser phase spaces, in problems on the combinatorics  of finite Weyl groups and Iwahori-Hecke algebras (quasi-invariants, Kazhdan-Lusztig cell theory). I will also show how this generalizes to finite complex reflection groups, often conjecturally.

Christian Korff: Cylindric Macdonald functions and a deformation of the Verlinde algebra
We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux, which are periodic continuations of ordinary skew tableaux, employing an integrable statistical lattice model on non-intersecting paths. We show that the cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which can be interpreted as a one-parameter deformation of the sl(n) Verlinde algebra, i.e. the structure constants of the Frobenius algebra are polynomials in a variable t whose constant terms are the Wess-Zumino-Novikov-Witten fusion coefficients. The latter are known to coincide with dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Alternatively, the deformed Verlinde algebra can be realised as a commutative subalgebra in the endomorphisms over a Kirillov-Reshetikhin module of the quantum affine sl(n) algebra. Acting with special elements of this subalgebra on a highest weight vector, one obtains Lusztig's canonical basis.

Andrey Mironov: Commuting differential operators with polynomial coefficients
We give examples of commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of arbitrary genus.

Alexei Penskoi: Extremal metrics: recent developments
An eigenvalue of the Laplace-Beltrami operator on a compact surface could be considered as a functional on the space of Riemannian metrics of fixed volume. The question about metrics extremal for this functional goes back to 70-80's pioneering works by Hersch, Yau et al and turns out to be very difficult. Recent developments in this area show an interesting interplay between minimal surfaces, classical equations of mathematical physics and group actions.

Leonid Rybnikov: Laumon spaces and Yangians
The talk is based on the joint work with A. Braverman, B. Feigin and M. Finkelberg. Laumon spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety. We construct a (type A) Yangian action in the localized equivariant cohomology of Laumon spaces by certain natural correspondences.
The same construction for partial flag varieties gives a geometric representation of the shifted (Brundan-Kleshchev) Yangian, satisfying some remarkable properties. This can be regarded as a "finite" version of the AGT conjecture.

Oleg Sheinman: Lax equations and Knizhnik-Zamolodchikov connection
In the talk, given a Lax integrable system with the spectral parameter on a Riemann surface, we construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields. For the Lax equations in question, we propose a way to represent Hamiltonian vector fields by covariant derivatives with respect to the (high-genus) Knizhnik-Zamolodchikov connection. This provides a pre-quantization of the Lax system.

Sergei Tabachnikov: Higher pentagram maps and cluster algebras
In the first talk, I shall introduce the pentagram map and explain why it is completely integrable (the integrals and the invariant Poisson bracket will be described). I shall consider the continuous limit of the pentagram map and show that it can be identified with the Boussinesq equation. I shall also describe new configuration theorems of projective geometry inspired by the pentagram map. I plan to demonstrate computer programs illustrating these results.
In the second talk, I shall describe a family of completely integrable systems generalizing the pentagram map. These rational maps are particular cases of cluster dynamics, understood in the framework of weighted directed networks on the torus. I shall introduce these subjects and deduce complete integrability of the maps. The geometric meaning of the maps will be explained as well.

Richard Thomas: Curves on algebraic surfaces
I’ll discuss the counting of holomorphic curves on complex algebraic surfaces in two ways: a modified (“reduced”) Gromov-Witten theory, and a modified theory of stable pairs. The latter can be computed completed in terms of various topological numbers. The results give rise to strange formulae involving modular forms.

Alexander Varchenko: Bethe algebras and geometric Langlands correspondence