Speaker: Thomas Walpuski (HU Berlin)

Title: Gauge theory and exceptional holonomy: progress and open problems

Abstract:
In the late 1990s, mathematicians started to (somewhat) seriously investigate gauge theory in higher dimensions, in particular on Riemannian manifolds with exceptional holonomy: G2 and Spin(7). The initial hope was that this might lead to higher-dimensional analogues of the Casson invariant, instanton Floer homology, etc.—which have turned out to be essential tools in the study of low-dimensional topology. Naive versions of this hope have been shattered. Nevertheless, in the quest to realise a more sophisticated version of this hope, mathematicians have discovered links between gauge theory in higher dimensions, calibrated submanifolds, (new) gauge theories in low dimensions, etc. and stumbled upon numerous open problems (mostly in geometric analysis). The purpose of this talk is to sketch some of these developments, and to explain what is known and what is not.