## Talks

The talks will be by invitation only. The speakers are:

- A. Bobenko (TU Berlin, Germany)
- A. Bolsinov (Loughborough U., UK)
- F. Burstall (U. of Bath, UK)
- M. Dunajski (U. of Cambridge, UK)
- V. Gritsenko (U. of Lille, France/NRU HSE, Russia)
- B. Konopelchenko (INFN, Section of Lecce, Italy)
- S. Lombardo (Loughborough U., UK)
- P. Lorenzoni (U. of Milano Bicocca)
- V. Sokolov (Landau Inst. of Theor. Phys. of RAS, Russia)
- I. Strachan (U. of Glasgow, UK)

## Program

**All time is UK local time**.

10:00 - 10:05 | Welcome | |

10:10 - 11:00 | I. Strachan | DT invariants, integrability and hyper-Kahler metrics |

11:10 - 12:00 | V. Sokolov | Non-Abelian systems with conservation laws and symmetries |

12:10 - 13:00 | M. Dunajski | Null-Kähler geometry and integrability |

Lunch break | ||

14:00 - 14:50 | A. Bobenko | Orthogonal ring patterns and integrable systems |

15:00 - 15:50 | P. Lorenzoni | Flat F-manifolds, Riemannian F-manifolds and integrable hierarchies |

16:00 - 16:50 | V. Gritsenko | Polynomial rings of Jacobi forms and non-linear differential equations |

Tea break | ||

18:00 | Conference party | Foreword by Alexander Veselov |

10:00 - 10:50 | A. Bolsinov | Nijenhuis Geometry: gl-regular Nijenhuis operators |

11:00 - 11:50 | F. Burstall | On n-soliton isothermic surfaces in closed form |

Coffee break | ||

12:10 - 13:00 | S. Lombardo | Automorphic Lie algebras of modular type |

13:10 - 14:00 | B. Konopelchenko | Self-dual Einstein spaces, general heavenly and TED equations |

14:00 | Conclusion | Afterword by E. Ferapontov |

## List of abstracts

Alexander Bobenko (Technische Universität Berlin) *Orthogonal ring patterns and integrable systems.* **Abstract:** We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that they are governed by discrete integrable equations. We construct ring patterns analogues of the Doyle spiral, Erf and z^α functions. We also derive a vibrational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions. Relation to discrete surfaces with constant mean curvature is established. This is a joint work with T.Hoffmann and T.Roerig.

Alexey Bolsinov (University of Loughborough): *Nijenhuis Geometry: gl-regular Nijenhuis operators*. **Abstract:** We study Nijenhuis operators, that is, (1,1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., belong to regular orbits of the natural gl-action. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local description of such operators. We apply this local description to study singular points. In particular, we obtain their normal forms in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces.

Francis Burstall (University of Bath) *On n-soliton isothermic surfaces in closed form.* **Abstract:** I will describe a closed form formula for the isothermic surfaces in a Bianchi n-cube of iterated Darboux transforms. The formula is valid in the smooth, discrete and semi-discrete settings but is inspired by an application of the vectorial Ribaucour transform of Dajczer-Florit-Tojeiro and Liu-Manas in the smooth case.

Maciej Dunajski (University of Cambridge) *Null-Kähler geometry and integrability.* **Abstract:** We construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear in the Bridgeland stability conditions of the moduli spaces of Calabi-Yau three-folds. Using twistor methods we show that, in dimension four - where there is a connection with dispersionless integrability - the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.

Valery Gritsenko (University of Lille and NRU HSE Moscow) *Polynomial rings of Jacobi forms and non-linear differential equations.* **Abstract:** We construct a tower of generators of the polynomial rings of the weak Jacobi modular forms invariant with respect to the Weyl group W(D_n) for all n>1. For the root lattice D_n there are two types of generators. The most important generators of index one are related to Lorentzian Kac-Moody algebras, the BCOV-analytic torsions and, hypothetically, with Gromov-Witten invariants. In this talk, we discuss the differential equations that these generators satisfy. The results will be published soon in our joint preprint with Dmitry Adler.

Boris Konopelchenko (Istituto Nazionale di Fisica Nucleare sezione di Lecce) *Self-dual Einstein spaces, general heavenly and TED equations.* **Abstract: ** Several aspects of interrelations between the self-dual Einstein spaces, heavenly type equations and TED equation are discussed. Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the general heavenly equation being the underlying governing equation. The formalism developed is used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebanski's first and second heavenly equations is retrived and interpreted in terms of eigenfunctions. As a particular application, it is proved that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the metrics do not admit any conformal Killing vectors. In addition, connections with travelling wave reductions of the TED equation which constitutes a 4+4-dimensional generalization of the general heavenly equation are found. Symmetries of the TED equation and its relevance to the Kähler geometry, in particular, the Fubini-Study metric are discussed. The is talk based on the recent joint paper with W.Schief and A.Szereszewski: Self-dual Einstein spaces and general heavenly equation. Eigenfunctions as coordinates, arXiv:2008.07261.

Sara Lombardo (University of Loughborough) *Automorphic Lie algebras of modular type.* **Abstract:**We introduce and study hyperbolic versions of automorphic Lie algebras for the modular group SL(2,Z) acting naturally on the upper half-plane and by conjugation on any simple finite-dimensional complex Lie algebra.

Paolo Lorenzoni (University of Milano Bicocca) *Flat F-manifolds, Riemannian F-manifolds and integrable hierarchies.* **Abstract:** We discuss integrable hierarchies associated with flat F-manifolds. Based on joint works with A. Arsie, A. Buryak and P. Rossi.

Vladimir Sokolov (Landau Institute for Theoretical Physics Chernogolovka and UFABC Sao Paulo) *Non-abelian systems with conservation laws and symmetries.* **Abstract:** We find noncommutative analogs for well-known polynomial integrable systems of ODEs and PDEs with two unknown variables. Based on joint papers with V.Adler and T.Wolf: 1. V.V.Sokolov and T.Wolf, Non-commutative generalization of integrable quadratic ODE-systems, Letters in Mathematical Physics 110 no 3, 533--553 (2020); arXiv:1807.05583 2. V.E.Adler and V.V.Sokolov, Nonabelian evolution systems with conservation laws, submitted to: Journal of differential equations; arXiv:2008.09174 (2020)

Ian Strachan (University of Glasgow) *DT invariants, integrability, and hyper-Kähler metrics.* **Abstract:** Hyper-Kähler metrics provide examples of 4N-dimensional integrable systems. In this talk the geometry and integrability of such systems will be reviewed, together with the connection to the geometry of Donaldson-Thomas invariants, via a certain class of Riemann-Hilbert problems. The connection between Moyal deformations of these hyper-Kähler metrics and quantum DT invariants will also be explained. This is a joint work with Tom Bridgeland.