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Title: Yang-Mills theory and random surfaces Abstract: The Clay Institute famously offered one million dollars to anyone who could mathematically construct and understand a certain continuum version of “Yang-Mills gauge theory.” This theory is the basis of the standard model of physics, and the heart of the problem is to understand the so-called “Wilson loop expectations.” Following recent work with Sky Cao and Minjae Park, I will explain how the “Wilson loop expectations” in a lattice Yang-Mills model are equivalent to “insertion costs” of loops in a related random-closed-surface-ensemble model. In a sense, these results allow us to convert one famously hard problem into another presumably hard problem. But the new problem is all about random surfaces and random permutations, and it has a lot of relationships with and similarities to other problems we understand (think domino tilings, random planar maps, Young tableaux and symmetric group representation theory, and the Weingarten calculus). It gives us some intuition for *why* certain things should be true like the “area law” or “exponential correlation decay” (what physicists call “quark confinement” or “mass gap”) even if we can’t prove all of them yet.