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Joel Fine: Lectures

September 13, 2017
TITLE: Hypersymplectic 4-manifolds and the G2 Laplacian flow

ABSTRACT:The G_2 Laplacian flow (introduced independently by Bryant and Hitchin) is designed to deform a closed G_2 structure to one with vanishing torsion. I will describe joint work with Chengjian Yao on this flow in the special case when the underlying 7-manifold is the product of a 4-manifold X and a 3-torus and the G_2 structure is torus invariant. The G_2 structure can be described in terms of a triple of symplectic forms on X, called a hypersymplectic structure, and this leads to a problem in 4-dimensional symplectic topology which is of interest in it’s own right. Our main result is that in this situation the G_2 Laplacian flow can be extended for as long as the scalar curvature remains bounded. The proof involves ideas from Ricci flow, but it is interesting that the result is stronger than what is currently known for the 4-dimensional Ricci flow. The proof exploits the hypersymplectic structure to carry out a more refined blow-up analysis than is directly possible in the Ricci flow case.

January 9, 2017
TITLE: Adiabatic limits and constant scalar curvature Kähler metrics

ABSTRACT: I will describe how to use adiabatic limits to find constant scalar curvature Kähler metrics, emphasising aspects which carry over to adiabatic limits in general. I will focus on the cases when all fibres are non-singular, as these are completely understood (due to Hong, Brönnle and also my PhD thesis).