Geometry and Analysis Seminar

The moduli space of Higgs bundles on a compact Riemann surface C for a group G is diffeomorphic to the character variety of representations 

of the fundamental group in G. One description depends on the complex structure of C, the other is purely topological. Using a natural symplectic Ehresmann connection we show how to build the complex structure on the family of Higgs bundle moduli spaces over Teichmuller space and derive some consequences for the energy of the associated harmonic maps.

Lagrangian mean curvature flow (LMCF) is a way to deform a Lagrangian submanifold inside a Calabi--Yau manifold according to the negative gradient of the area functional. There are influential conjectures about LMCF due to Thomas--Yau and Joyce, describing the long-time behaviour and singularities of the flow. By foundational work of Neves, Type I singularities are ruled out under mild assumptions, so it is important to construct examples of Type II singularities with a given blow-up model. In this talk, we describe a general method to construct examples of Lawlor neck pinching in LMCF in complex dimension at least 3. We employ a P.D.E. based approach to solve the problem, as an example of 'parabolic gluing'. The main technical tool we use is the notion of manifolds with corners and a-corners, as introduced by Joyce following earlier work of Melrose. Time permitting, we will discuss how one may construct examples of generic neck pinching.