Low-dimensional topology is the study of manifolds and cell complexes in dimensions four and below. Input from geometry and analysis has been central to progress in this field over the past four decades, and this article will focus on one aspect of these developments in particular, namely the use of Yang-Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds, but the past ten years have seen new applications of gauge theory, and new interactions with more recent threads in the subject, particularly in 3-dimensional topology. This is a field where many mathematical techniques have found applications, and sometimes a theorem has two or more independent proofs, drawing on more than one of these techniques. We will focus primarily on some questions and results where gauge theory plays a special role.
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