Hi! This month, I thought I would start a brief series of articles describing the uses of gauge theory in mathematics. Rather than discuss current research directions in gauge theory (of which there are many), I hope to give an overview of the sorts of mathematical questions that gauge theory was first used to answer and a general idea of what it is all about. Our goal for today will be to contextualize the initial advances in low-dimensional topology due to gauge theory by giving a picture of the state of affairs before its introduction. We will thus spend this post establishing some basic terms and ideas for the uninitiated; we will wait until later posts to discuss gauge theory itself and what it can tell us about topology.
I have attempted to make this article as introductory and motivational as possible, especially to readers who are less familiar with the finer historical developments in low-dimensional topology. No background is needed except for (at best) a passing recollection of basic algebraic and differential topology. In several places I have been a bit cavalier with precise definitions for the sake of the exposition. All errors are (of course) mine!