Georgia Institute of Technology College of Sciences

Gauge theory is a beautiful subject that studies the space of connections on a vector bundle. It is also the natural language in which theories of particle physics are formulated. In fact, the word "gauge" has its origins in electromagnetism, and in this talk, we explore the basic geometric objects of gauge theory and show how one explicitly recovers the classical Maxwell's equations as a special case of the equations of gauge theory . Next, generalizing Maxwell's equations to a ``nonabelian" setting, we obtain the Yang-Mills equations, which describe the electroweak force in nature. Surprisingly, these equations were used by Simon Donaldson in the 1980s to prove spectacular results for the topology of smooth four-manifolds. We conclude this talk by describing some of the beautiful geometry and analysis behind gauge theory that goes into the work of Donaldson (for which we awarded a Fields Medal), and time permitting, we hope to say a bit about other gauge-theoretic applications to low-dimensional topology, for instance, instanton Floer homology.