Research Seminar

Title: Higher homotopy groups of G₂-moduli spaces

Abstract:  Riemannian manifolds M with holonomy equal to the exceptional Lie group G₂ are difficult to construct.  Joyce proved both that compact examples exist and that their moduli spaces (meaning we mod out by diffeomorphism isotopic to the identity) are always locally Euclidean (of dimension the third Betti number of M).

Despite Joyce’s results, very little is known about the global topology of G₂-moduli spaces, and until recently it was not known whether any path component of a G₂-moduli space was not contractible.  I will report on joint work with Sebastian Goette and Thorsten Hertl, where we exhibit components of some G₂-moduli spaces, which have non-trivial second homotopy groups.

Our construction of non-trivial S²-families of G₂-manifols involves a parametrised version of Joyce’s original “Kummer construction” of G₂-manifolds by resolution of singularities and also leads to related non-zero classes of π₂ BDiff(M).

Gibbons-Hawking.

Minimal submanifolds.

Title: The Moduli space of Ricci-flat metrics on a K3 surface.

Abstract: It is known that if M is a compact spin manifold, then the moduli
space of Ricci-flat metrics modulo the action of the diffeomorphism group
is a smooth manifold of finite dimension. In this talk, we consider M to be a
carefully constructed K3 surface and outline an argument showing that in
this case, the moduli space is not complete. 

Title: A class of locally inhomogeneous complete quaternionic Kähler manifolds.

Abstract: We prove that the one-loop deformation of any quaternionic Kähler manifold
in the class of c-map spaces is locally inhomogeneous. As a corollary, we obtain that the
full isometry group of the one-loop deformation of any homogeneous c-map space has
precisely cohomogeneity one.
This is a joint work with Vicente Cortés and Arpan Saha (https://arxiv.org/abs/2210.10097).

The Stenzel metric on Tangent Bundle of Spheres.

Calabi’s two families of examples.

Monge-Ampe`re on ALE spaces.

Finding Calabi-Yau metrics - the Monge-Ampe`re route.

More Ka ̈hler geometry and special holonomy.

Basics of (hyper-)Kähler geometry and special holonomy.

Notation, reviewing the basics:

 connections on bundles, pullback connections, etc...

Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

Max Schult
(Leibniz Universität Hannover)

Andreas Savas-Halilaj
(University of Ioannina)

Lothar Schiemanowski
(Leibniz Universität Hannover)

Balázs Békési
(Leibniz Universität Hannover)

 tba

Complete families of embedded high genus CMC surfaces in the 3-sphere