Abstract: One way to interpret the fundamental theorem of calculus is that the difference in a real-valued smooth function across an interval is controlled by the integral of its derivative on that interval. In a similar way, a classical Poincaré inequality states that for a smooth function on a ball in Euclidean space, the L^p norm of the deviation of the function from its average is controlled by the L^p norm of the gradient of the function. In recent years analogous inequalities have been studied on general metric spaces, where they may or may not hold depending on the particular space.
In this talk, I'll discuss work with Hume and Tessera on Poincaré profiles which measure, on large scales, the extent to which such inequalities hold. This idea generalises the Separation Profile of Benjamini, Schramm and Timar. I'll survey some of the history and motivations of these ideas, applications to coarse embedding problems, and situations where an interesting dichotomy between analytically thin and analytically thick groups is seen. Joint with Hume and Tessera.