Programme
Boris Kruglikov: Dispersionless integrable hierarchies and GL(2) geometry
Abstract: Paraconformal or GL(2) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree (n-1) in the projectivized cotangent (or tangent) bundle PT*M. Such cone structures arise in Poisson geometry, exotic holonomy, algebraic geometry, ODE theory, Grassmann geometry. It was studied, in particular, by Bryant, Nurowski, Tod, Dunajski, Krynski. In particular, in the works of latter, this geometry was presented as a natural generalization of the Einstein-Weyl geometry. This is the background geometry in 3D for dispersionless second order PDEs. We discuss how GL(2) structures in general dimension arise through integrable hierarchies of integrable PDEs in 3D, and how general such structures can be encoded via an integrable “master” PDE. The work is joint with Eugene Ferapontov.
Wojciech Kryński: On deformations of integrable systems
Abstract: We discuss a method of obtaining new integrable systems from a given one by equipping the corresponding twistor space with appropriate additional structure. Examples include: the dispersionless Hirota equation and the generalized heavenly equation.
Omid Makhmali: Half-flatness in dimension 4: from conformal to causal structures
Abstract: In conformal geometry of neutral signature, half-flatness can be interpreted as an Frobenius integrability condition on the sky bundle of the conformal structure. This suggests an immediate generalization of such a notion to the broader class of causal structures where the null cones do not have to be quadratic. We focus on the case where the null cones are cones over ruled Cayley's cubic and discuss the similarities and challenges that arise compared to the classical conformal setting.
Andriy Panasyuk: Introduction to twistor spaces and some problems in classification of gravitons
Abstract: In a recent paper on the so-called general heavenly equation, which describes self-dual vacuum Einstein metrics, Konopelchenko, Schief and Szereszewski observed that any solution of the Hirota dispersionless system in 4-D also solves this equation. Geometrically this means that 2-dimensional Kronecker web standing behind the heavenly equation is included in a 3-dimensional Veronese web described by the Hirota system. Natural problem arises: to describe those solutions of the heavenly equation which come from Veronese or other webs of dimension 3. I will try to reformulate this problems in terms of twistor spaces related to Kronecker and Veronese webs.
Vsevolod Shevchishin: Webs parametrised by elliptic curves
Abstract: Webs are families of holomorphic foliations (F_s, s\in S), on a given complex manifold M parametrised by a given compact complex manifold S. In the classical theory of webs the most studied case is when the parameter space is a rational curve S = \CP^1. Those are, for example, Veronese webs when codim F_s =1, and Kronecker webs when codim F_s =2. In my talk I discuss some basic properties of webs parameterised by an elliptic curve, i.e. compact complex curve of genus g(S)=1.
Adam Szereszewski: Schief equation
Abstract: Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces. Developed method is used to link a variety of known heavenly equations (Plebański, Husain-Park, Schief equations). The reduction of heavenly equations is also found.