What's Happening in Fine Hall

The spectral gap of a geometric object---think of a graph, a manifold, a group---measures, for example, how quickly a random walker explores the entire object. It is therefore of special interest to contruct objects with the largest possible spectral gap. Are such objects rare or ubiquitous? In many cases, it is conjectured that optimal spectral gaps are generic, but this has been challenging to prove. I will describe a new approach developed over the past two years that made it possible to establish such results in many previously inaccessible situations.