Workshop

In this talk I will review the emergence of Yang-Mills theories and other related gauge theories (including Seiberg-Witten theories and Hitchin Systems) as they arise in modern string theory constructions. I will demonstrate the way that string dualities can relate seemingly disparate geometric systems and shed light on novel underlying mathematical structure. As one example, I will focus on generalizations of Hitchin systems and Higgs bundles as they appear in compactifications of F-theory.

The purpose of this talk is to explain why, despite severe analytical difficulties, it might be worthwhile to study gauge theory in higher dimensions after all.

In these two talks I will introduce the basic elements of mathematical gauge theory, including connections, gauge transformations, curvature, the Yang-Mills functional, instantons, and Uhlenbeck's Theorems.

We will discuss the Hermitian-Yang-Mills equation and the Donaldson-Uhlenbeck-Yau theorem relating the existence of Hermitian-Yang-Mills connections to stability of holomorphic vector bundles. Depending on time we will also discuss more recent developments in this area. 

Instanton Floer homology groups are groups associated to a 3-manifold. Their construction involves a process analogous to Morse theory where we consider a functional on the space of SU(2) connections on the manifold and study its critical points and gradient flow lines. In these two talks, we'll define the instanton Floer homology and discuss how to use it to understand 3-manifolds, knots, and links.

In their seminal article published in 1998, Donaldson and Thomas proposed the study of gauge theory in higher dimensions, which has seen significant interest in recent years.  This theory requires manifolds of dimensions 6, 7 and 8 endowed with geometric structures that induce Riemannian metrics with special holonomy.  In this minicourse I will first describe the basics of these manifolds with special holonomy and gauge theory in this setting, before detailing some of the key questions and challenges in the field.  I will also discuss a range of the results which have been obtained in gauge theory and special holonomy to give a flavour for recent progress and open problems.

In this talk we will introduce Higgs bundles and generalized hyperpolygons, and look into different directions that have attracted attention within the area in the last decade.  In particular, we shall see how Mirror Symmetry can be seen in terms of branes within the Hitchin fibration and show that, under certain assumptions on flag types for generalized hyperpolygons, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we will also see how I have been using my geometric background to answer questions in other areas of science. Much of the talk follows recent work joint with Steven Rayan.

In their seminal article published in 1998, Donaldson and Thomas proposed the study of gauge theory in higher dimensions, which has seen significant interest in recent years.  This theory requires manifolds of dimensions 6, 7 and 8 endowed with geometric structures that induce Riemannian metrics with special holonomy.  In this minicourse I will first describe the basics of these manifolds with special holonomy and gauge theory in this setting, before detailing some of the key questions and challenges in the field.  I will also discuss a range of the results which have been obtained in gauge theory and special holonomy to give a flavour for recent progress and open problems.

We will discuss the Hermitian-Yang-Mills equation and the Donaldson-Uhlenbeck-Yau theorem relating the existence of Hermitian-Yang-Mills connections to stability of holomorphic vector bundles. Depending on time we will also discuss more recent developments in this area.