Hausdorff Kolloquium 2017/18

Date: November 22, 2017 - January 24, 2018

Venue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn


Mark Gross (Cambridge): A general mirror symmetry construction

Mirror symmetry is an area of geometry which had its beginnings in string theory around 1989, and has led to a great deal of interesting mathematics. It was noticed in many examples that certain kinds of complex manifolds called Calabi-Yau manifolds came in pairs, and there was a subtle duality between the geometry of the members of the pairs, allowing for previously inaccessible algebro-geometric calculations to be carried out. I will survey some of the main ideas of mirror symmetry since its origins around 1990, leading to recent work joint with Siebert giving a general construction of mirror pairs.


Frances Kirwan (Oxford): Moduli spaces of unstable curves

Moduli spaces arise naturally in classification problems in geometry. The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s.  Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. The aim of this talk is to describe these moduli spaces and outline their GIT construction, and then explain how recent methods from non-reductive GIT can help us try to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves (of fixed singularity type).


Andrea Malchiodi (Pisa): Variational structure of Liouville equations

Liouville equations arise when trying to uniformize curvatures or to extremise energies depending on spectra of surfaces. We consider cases when Gauss-Bonnet integrals are "large", as it might happen in presence of conical singularities. We prove existence of solutions combining geometric functional inequalities and a micro/macroscopic study of conformal volume distribution.