Typical hyperbolic surfaces have an optimal spectral gap

Add to your list(s) Download to your calendar using vCal

• Laura Monk (Bristol)
• Wednesday 28 May 2025, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Oscar Randal-Williams.

The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus g, equivalently), we know that the spectral gap is asymptotically bounded above by 1/4. The aim of this talk is to present joint work with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any ε>0, the Weil-Petersson probability for a hyperbolic surface of genus g to have a spectral gap greater than 1/4-ε goes to one as g goes to infinity. This statement is analogous to Alon’s 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman’s work, and introduce new tools and ideas that we have developed in order to tackle this problem.

This talk is part of the Differential Geometry and Topology Seminar series.

This talk is included in these lists:

• All CMS events
• All Talks (aka the CURE list)
• CMS Events
• DPMMS Lists
• DPMMS Pure Maths Seminar
• DPMMS info aggregator
• DPMMS lists
• Differential Geometry and Topology Seminar
• Hanchen DaDaDash
• Interested Talks
• MR13
• School of Physical Sciences
• bld31

Note that ex-directory lists are not shown.