Geometry and Topology Seminar

In the first part of this talk I will give a gentle introduction to G2-manifolds and discuss what I consider to be the major milestones in the last 20 years. In particular, I will review the techniques used to construct compact examples: Joyce's generalized Kummer construction and the twisted connected sum construction pioneered by Kovalev and extended and improved by Kovalev-Lee and Corti-Haskins-Nordström-Pacini. 

The second part of this talk revolves around Donaldson-Thomas's program to use gauge theory in order to construct enumerative invariants of G2-manifolds. I will talk about joint work with Henrique Sá Earp, which opens up ways to investigate this program on twisted connected sums. In particular, I will discuss concrete examples of G2-instantons discovered by myself and Menet–Nordström–Sá Earp (and give an interpretation of the later via the Fourier–Mukai transform). Finally, I will talk about my current understanding of the Donaldson–Thomas program, mention the difficulties that arise, and explain joint work with Andriy Haydys in which we propose using generalized Seiberg-Witten equations to overcome part of those complications.