Spectral gap on random hyperbolic surfaces

For a closed hyperbolic surface, the spectrum of its Laplacian on L^2 functions provides a plethora of information about its geometric structure. In this talk I will discuss the behaviour of the first non-zero eigenvalue, or spectral gap, of a surface. In particular, I will cover recent joint work with Will Hide (Oxford) and Davide Macera (Bonn) where we study the size of this spectral gap for uniformly random covers of a fixed base surface. This work employs a recent breakthrough of Magee, Puder and van Handel on strong convergence of surfaces groups and leads us to obtain polynomial error bounds for the spectral gap.