Modern Mathematics Lecture Series | Convergence of unitary representations and spectral gaps of manifolds

Abstract

Let M be a locally symmetric space. I'll discuss the notion of `strong convergence' of a sequence of finite dimensional unitary representations of the fundamental group of M. Once this convergence property is established for particular sequences of representations, it can be used to deduce information about the spectral gap of the Laplacian on covering spaces of M, or on vector bundles over M. This has led to several recent advances. For example, it is now known that every compact hyperbolic surface has a sequence of covering spaces with asymptotically optimal relative spectral gap. I'll discuss what is known and conjectured for higher dimensional hyperbolic manifolds.

Based on joint works with W. Hide, L. Louder, J. Thomas.

About the speaker

M. Magee obtained his Ph.D. in 2014 from U.C. Santa Cruz. After postdoctoral positions at IAS Princeton and Yale, he moved to Durham University where he has been a Professor since 2021. He is the winner of a Whitehead Prize (2021) and Philip Leverhulme prize (2023). In academic year 23-24 he is a von Neumann Fellow at IAS Princeton.