Moon Duchin: Fine asymptotic geometry

I will discuss work that studies the geometry of groups and spaces through random sampling against a measure, surveying results of this kind in finitely-generated groups (abelian, lamplighter, Heisenberg) and in the spaces associated to surface topology (curve graph, Teichmüller space). In the group setting, we develop tools for studying asymptotic density and other counting problems, which necessarily involve abandoning quasi-isometry invariance and studying finer large-scale statistics. These investigations lead to surprising connections with other kinds of mathematics. I'll touch on joint work with Lelièvre and Mooney, and with Dowdall and Masur.