Title: Gauge theory in generalised Hopf bundles

Abstract: I will present recent work on instantons and Yang-Mills connections in generalized Hopf bundles. Starting with the 7-sphere as a guiding example, we will explore key ideas and techniques for studying connections on these bundles.

Title: Kapustin-Witten Floer Theory and flat SL(2;C) connections on 3-manifolds

Abstract: The Kapustin-Witten equations may be able to be used to construct new gauge theoretic invariants in a similar way to Floer theory. To construct an instanton Floer style theory using these equations, one needs to unsderstand the space of flat SL(2; C) connections on three manifolds. The non-compactness of this space, arising from the non-compactness of SL(2; C), is problematic when trying to develop the theory. I will outline some potential methods for overcoming these issues so that the invariants may be defined.

Title: The moduli space of conically singular SU(3)-instantons

Abstract: SU(3)-instantons are distinguished connections on principal bundles over Calabi–Yau 3-folds (or, more generally, 6-dimensional manifolds equipped with an SU(3)-structure). The moduli space of these instantons is typically non-compact, with one possible source of non-compactness arising from the formation of non-removable singularities. In this talk, I will focus on instantons with isolated, non-removable singularities of conical type. In particular, I will discuss their moduli space and its structure. This talk is based on ongoing work with Yuanqi Wang.

Title: The heterotic SU(3) moduli space

Abstract: The heterotic SU(3) system arises from theoretical physics and is a coupled set of PDEs on a complex 3-fold intertwining differential forms on the manifold and gauge theoretic data on bundles over it. Perhaps the two most fundamental questions concerning the moduli space of solutions are: when is it non-empty and, if it is non-empty, what is its (expected) dimension? I will give an introduction to this system and describe what we know about these questions, including some recent joint work with de Lazari, Sa Earp and Svanes on the dimension question.