I am currently seeking PhD students to work on topics of the project

`Convergence of unitary representations'

Positions are fully funded for four years with generous travel allowance.
Research questions of this project involve subsets of Group Theory, Operator Algebras, Spectral Geometry, and Random Matrix Theory.
If you are interested in working with me, please get in touch.

ABRIDGED ABSTRACT.
Understanding the unitary representations of a given group G is one of the most persistent problems of mathematics. If G is the integers, the ensuing theory is that of the Fourier transform. If G is the Heisenberg group, then the resulting representation theory is the theory of matrix models of Quantum Mechanics. Accelerating, if G is the absolute Galois group of the rationals, the theory is described by the Langlands program.
My goal is to understand the finite dimensional (f.d.) unitary representations of discrete groups through the lens of how they can converge to the regular representation. I describe both weak and strong forms of convergence and focus mainly on strong convergence.
I first ask which groups have f.d. unitary representations that strongly converge to their regular representation? What if we require representations to factor through permutation groups? These questions 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 we have a way to randomize representations. In particular this applies to the fundamental groups of closed 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 have as yet no explicit examples of the phenomenon that the proposal in centered on! We present an algebraic candidate that is intimately related to Selberg’s Eigenvalue Conjecture in automorphic forms.
Most of the above questions have spectacular consequences to spectral gaps of locally symmetric spaces, a connection that I discovered with Hide. We do not understand this outside special cases yet. I ambitiously aim to completely describe the connections between strong convergence of representations of a lattice and their induced representations of the ambient Lie group. Finally, we imagine what lies ‘beyond strong convergence’ and whether random matrix theory can take us there.