Convergence of unitary representations

Understanding the unitary representations of a given group G is one of the most persistent problemsof mathematics. If G is the integers, the ensuing theory is that of the Fourier transform. If G isthe Heisenberg group, then the resulting representation theory is the theory of matrix models ofQuantum Mechanics. Accelerating, if G is the absolute Galois group of the rationals, the theory isdescribed by the Langlands program.My goal is to understand the finite dimensional (f.d.) unitary representations of discrete groupsthrough the lens of how they can converge to the regular representation. I describe both weak andstrong forms of convergence and focus mainly on strong convergence. I first ask which groups have f.d. unitary representations that strongly converge to their regularrepresentation? What if we require representations to factor through permutation groups? Thesequestions are deep, wide-ranging, and push far beyond the state-of-the-art.Next I ask to what extent strong convergence of f.d. unitary representations is generic, when wehave a way to randomize representations. In particular this applies to the fundamental groups ofclosed surfaces, which are a test bed for the current program. In many cases randomization is the only tool we know to establish strong convergence, so we haveas yet no explicit examples of the phenomenon that the proposal in centered on! We present analgebraic candidate that is intimately related to Selberg’s Eigenvalue Conjecture in automorphicforms. Most of the above questions have spectacular consequences to spectral gaps of locally symmetricspaces, a connection that I discovered with Hide. We do not understand this outside special casesyet. I ambitiously aim to completely describe the connections between strong convergence of representationsof a lattice and their induced representations of the ambient Lie group.Finally, we imagine what lies beyond strong convergence and whether random matrix theory cantake us there.