Optimal Spectral Gaps | Richard M. Karp Distinguished Lecture

Ramon van Handel surveys recent progress in establishing optimal spectral gaps in geometric structures using the polynomial method.

Ramon van Handel (Princeton University) simons.berkeley.edu/talks/ramon-van-handel-princeton-univers… Probabilistic Phenomena on Graphs, Groups, and Manifolds The spectral gap of a geometric structure — think of a graph, a manifold, a group — measures how quickly a random walker explores the entire structure. It is a problem of great interest, both in pure and in applied mathematics, to construct geometric structures with the largest possible spectral gap. In many cases, it is conjectured that optimal spectral gaps are ubiquitous, but this has been challenging to prove. Ramon van Handel will survey some remarkable progress that has been achieved in this area in the past two years and will describe a new approach, the polynomial method, that has made it possible to establish optimal spectral gaps in various new situations. (Based in part on joint works with C.-F. Chen, J. Garza-Vargas, M. Magee, D. Puder, and J. Tropp.) Ramon van Handel is a Professor of Mathematics at Princeton University. He received his PhD in 2007 from Caltech. His research interests lie broadly in probability theory, analysis, geometry, and their interactions. His honors include the NSF Career and PECASE awards, the Erlang Prize, an ICBS Frontiers of Science Award, the Princeton University Graduate Mentoring Award, and several teaching awards. He was an invited speaker at the 2026 ICM, and is currently serving as co-Editor-in-Chief of the journal Probability Theory and Related Fields.