Heat Kernels, Plancherel Duality, and the Selberg Trace Formula

Two very different corners of mathematics turn out to share the same engine room. The Selberg trace formula connects the way a curved surface vibrates (its Laplacian eigen values) to the lengths of the closed loops you could walk on it (its geodesics). Completely separately, physicists solved two-dimensional Yang–Mills theory- a toy version of the force that binds quarks-exactly, using a sum over "representations" of asymmetry group. This paper shows that both results are the same trick, applied to two different kinds of symmetry group: a non-compact one (rotations of the hyperbolic plane) for Selberg, and a compact one (the gauge group, like SU(3)) for Yang–Mills. Both stories start from a heat-flow process on a group and a duality theorem (Harish-Chandra's for the first, Peter–Weyl's for the second) that swaps a geometric description for an algebraic one. We work out the dictionary between the two stories in full, check every step against real numerical computation, and are explicit about where the analogy is exact, where it is loose, and where it simply does notapply - most importantly, it says nothing new about the four-dimensional mass gap.