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SUMMARY:Typical hyperbolic surfaces have an optimal spectral gap - Laura M
 onk (Bristol)
DTSTART:20250528T150000Z
DTEND:20250528T160000Z
UID:TALK231118@talks.cam.ac.uk
CONTACT:Oscar Randal-Williams
DESCRIPTION:The first non-zero Laplace eigenvalue of a hyperbolic surface\
 , or its spectral gap\, measures how well-connected the surface is: surfac
 es with a large spectral gap are hard to cut in pieces\, have a small diam
 eter and fast mixing times. For large hyperbolic surfaces (of large area o
 r large genus g\, equivalently)\, we know that the spectral gap is asympto
 tically bounded above by 1/4. The aim of this talk is to present joint wor
 k 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 infin
 ity. This statement is analogous to Alon’s 1986 conjecture for regular g
 raphs\, proven by Friedman in 2003. I will present our approach\, which sh
 ares many similarities with Friedman’s work\, and introduce new tools an
 d ideas that we have developed in order to tackle this problem.\n\n
LOCATION:MR13
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