Seminar:
Group Actions and Dynamics
Event time:
Monday, February 21, 2022 - 4:00pm
Location:
Zoom
Speaker:
Michael Magee
Speaker affiliation:
Durham University
Event description:
A hyperbolic surface is a surface with metric of constant curvature -1. The spectral gap between
the first two eigenvalues of the Laplacian on a closed hyperbolic surface contains a good deal of
information about the surface, including its connectivity, dynamical properties of its geodesic flow,
and error terms in geodesic counting problems. For arithmetic hyperbolic surfaces the spectral gap
is also the subject of one of the biggest open problems in automorphic forms: Selberg’s eigenvalue
conjecture.
It was an open problem from the 1970s whether there exist a sequence of closed hyperbolic sur-
faces with genera tending to infinity and spectral gap tending to 1/4. (The value 1/4 here is the
asymptotically optimal one.) Recently we proved that this is indeed possible. I’ll discuss the very
interesting background of this problem in detail as well as some ideas of the proof. This is joint work
with Will Hide.