Yorkshire and Durham Geometry Day at Leeds

Geometry event

10.30 Coffee
Maths Common Room

11.30 Gerasim Kokarev (Leeds):
"Harmonic maps and extremal eigenvalue problems"
Abstract: I will discuss a non-local variational problem asking to find a metric of unit volume on a two-dimensional surface that maximises the first non-zero Laplace eigenvalue. I will describe a recent progress on this problem and survey a collection of related results due to a number of people. The important ingredient in this study is a relationship with harmonic maps into spheres.

12.30 Lunch
University Refectory

13.30 Felipe Contatto (Cambridge):
"Metrisability of Painleve equations and first integrals of affine connections"
Abstract: The metrisability problem posed by Roger Liouville in 1889 is the following: given a particular set of curves on a manifold, we can ask whether they are geodesics of a pseudo-Riemannian metric. We solve this problem locally for the integral curves of the Painleve equations and we conclude that they are geodesics of a metric only for special choice of parameters, those for which the Painleve equations admit a first integral quadratic in first derivatives. These first integrals can be derived from Killing vectors of the corresponding metrics, providing a geometrical interpretation to them. In the second part of the talk I will present the following problem: given a torsion free affine connection on a surface, when does it admit 0, 1, 2 or 3 Killing 1-forms? The necessary and sufficient conditions for the existence of Killing 1-forms are established by prolongation and Frobenius theorem. I will explain how one can interpret hamiltonian descriptions of hydrodynamic type systems as a particular case of the latter geometrical problem. Both problems of metrisability and existence of Killing forms are not unrelated and I will explain where they overlap.

14.30 Marina Logares (Oxford):
"Higgs bundles, Integrable systems, singularities and a Torelli theorem"
Abstract: We will introduce Higgs bundles over a punctured Riemann surface. The moduli space of such objects describes an integrable system that completely determines the Riemann surface together with the punctures. Hence, it provides a Torelli type theorem for such moduli spaces. This is joint work with I. Biswas and T. Gomez

15.30 Tea
Maths Common Room

16.15 Kai Zheng (Warwick):
"Constant scalar curvature Kaehler metrics with cone singularities"
Abstract: In this talk, we will present recent research on constant scalar curvature Kaehler (cscK) metrics with cone singularities. We will mainly discuss results on the uniqueness problem, and also the existence problem if time allows.

18.00 Dinner at a local restaurant