Gauge theory, from low dimensions to higher dimensions and back

Almost thirty years ago, Donaldson and Thomas proposed extending powerful ideas from gauge theory, which had transformed the study of three- and four-dimensional manifolds, to higher dimensions, with the goal of defining new invariants of special holonomy manifolds. In this talk, I will outline the main ideas behind this program, mention some recent progress, and describe the key obstacles that remain, most notably non-compactness phenomena that make the analysis difficult.