Dr Monk and Dr Nalini Anantharaman, Collège de France, released a preprint answering a well-sought-after conjecture related to the spectral gap of random hyperbolic surfaces.

The spectral gap of a geometric object is a number that quantifies its connectivity: the bigger the spectral gap, the more connected the object. For instance, dumbbells (objects with two components connected by a very narrow tube) are poorly connected and have a very small spectral gap. The objects studied in this work are compact hyperbolic surfaces, interesting due to their chaotic nature and their links with number theory. In the large-scale limit, we know that the spectral gap is bounded above by the number ¼. However, no family of expanding surfaces of spectral gap ≥¼ is currently known. The existence of such a family, conjectured in 1984 by Buser, was established in 2021 by Hide and Magee using a probabilistic construction. Anantharaman and Monk proved that these surfaces not only exist: they are typical, meaning that, for any ε>0,  the probability for a surface to have a spectral gap at least ¼-ε goes to one in the large-scale limit.

The proof is inspired by Friedman's breakthrough proof of the same result for regular graphs, the Alon conjecture, and relies on introducing entirely new methods to study the length-distribution of closed geodesics on random hyperbolic surfaces.

This is the conclusion of a seven-year-long project, involving 300 pages of mathematics. Quanta magazine has covered this advance in a beautiful article, tracing back the story to Fields medallist Mirzakhani's influential work.