We hope to provide training for PhD students, and anyone else who would like to join. Our current programme is below.
Henan University short course: November 2020 - May 2021
Nuno Romão - Moduli Spaces in Gauge Theory
This course will cover rigorous aspects of gauge theory - a branch of
differential geometry that has been a major stage for
cross-fertilisation between mathematics and physics over the last few
decades. The focus will be on mathematical objects called moduli
spaces that play a pivotal role in this subject. They arise as spaces
of orbits (in some set of "fields", in physical terminology) under
infinite-dimensional groups of symmetries.
The lectures will be organised in two blocks. In Part I, the language of connections in fibre bundles will be introduced from scratch, together with other topics that are essential to lay out the foundations of gauge theory. In Part II, we will apply this machinery to the specific setting of the vortex equations, which are a system of PDEs on complex manifolds that form a prototype of the intriguing notion of self-duality, permeating much of modern geometry. The moduli spaces that we shall mostly be concerned with classify (or parametrise) equivalence classes of solutions to the vortex equations. These spaces will be described concretely in some simple examples, incorporating recent research. After this, we will be able to ask and answer some questions regarding their intrinsic geometry and topology, and along the way discuss a few applications.
Part I: Introduction to gauge theory
The sessions (2h15m each, no breaks) will start at 9:30am GMT(UTC) on Tuesdays 24th November and 1st, 8th December 2020; and then on Monday 14th, Tuesday 22nd December 2020.
Part II: Moduli of symplectic vortices
The sessions (2h15m each, no breaks) will start at 7:00am GMT(UTC) on Wednesday 3rd March and continue every Wednesday for about 10 weeks.
If you are interested in attending this short course organized by Henan University, please contact n.m.romao (at) gmail (dot) com for details.