Abstract:

I will explain how a simple symmetry assumption (“gauge invariance") leads to the notions of a "covariant derivative” and “curvature”.  This will naturally lead us to Yang–Mills theory, a vast generalizations of Maxwell’s theory of electro-magnetism.  After a procedure called “gauge fixing”, the Yang–Mills equation becomes a system of elliptic partial differential equations.  As we will see, this PDE is critical in dimension four.

The study of spaces of special solutions of the Yang–Mills equation (called “instantons”) naturally leads to many interesting problems in differential geometry and partial differential equations.  One of the simplest classes of “instantons" are anti-self-dual Yang–Mills connections over 4–manifolds.  Theses solutions where discovered by physicists,  and their study has lead to spectacular applications to the topology of 4–manifolds.  I will briefly discuss one of those applications and possibly sketch the idea of the proof.

A further important class of “instantons” are Hermitian Yang–Mills connections over Kähler manifolds.  The study of those has deep links with complex/algebraic geometry: the work of Donaldson and Uhlenbeck–Yau establishes an equivalence between non-singular Hermitian Yang–Mills connections and µ–stable holomorphic vector bundles.  This correspondence has been extended by Bando–Siu to the setting of singular Hermitian Yang–Mills connections.  In the final part of the talk I will discuss recent work with Adam Jacob and Henrique Sá Earp on understanding the precise nature of isolated singularities in the Hermitian–Yang Mills connections constructed by Bando–Siu.