Kronecker Webs and Nonlinear PDEs

A web on a manifold is a collection of foliations of constant dimension in a general position. Theory of classical webs, i.e. finite collections of foliations, was developed in the first half of XX century by the school of W. Blaschke. In the end of that century I. Gelfand and I. Zakharevich introduced Kronecker webs (Gelfand et al., J Funct Anal 99:150–178, 1991), collections of foliations parametrized by one-dimensional projective space with a particular dependence on the parameter, as objects encoding the local geometry of finite-dimensional bihamiltonian structures, i.e. pairs of compatible Poisson structures. Later Kronecker webs appeared as independent object of investigations in relation with integrable nonlinear systems of PDEs (Zakharevich 2000; Dunajski et al. Math Proc Camb Philos Soc 157:139–150, 2014; Kruglikov and Panasyuk, J Geom Phys 115:45–60, 2017; Panasyuk, Banach Center Publ 117:177–210, 2019). In this series of lectures I will give an outline of this relation based on the last two references and also explain recent ideas putting preceding results into the context of the so-called heavenly PDEs describing self-dual vacuum Einstein metrics in neutral signature (Panasyuk and Szereszewski, Class Quantum Grav 40:235003, 2023).