Explore the latest advancements in spectral geometry research through this seminar talk delivered by Laura Monk from the University of Bristol. Delve into the fascinating world of hyperbolic surfaces and their spectral gaps, which provide insights into surface connectivity, diameter, and mixing times. Learn about the ongoing project with Nalini Anantharaman, aiming to prove that most hyperbolic surfaces possess a near-optimal spectral gap. Discover the similarities between this approach and Alon's 1986 conjecture for regular graphs, proven by Friedman in 2003. Gain an understanding of the trace method employed, the challenges encountered, and the new tools developed to overcome them. Examine the current progress, which has improved the spectral gap bound from 3/16 - ε to 2/9 - ε, surpassing previous results by Wu-Xue and Lipnowski-Wright. Conclude by discussing the final steps required to achieve the optimal result in this cutting-edge mathematical research.