It was shown by Balasubramanya that any acylindrically hyperbolic group (a natural generalisation of a hyperbolic group) must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concepts of acylindrically hyperbolic and acylindrically arboreal groups and discuss some particularly interesting examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees.

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov-Hausdorff distance, by finite dimensional Banach spaces. Time permitting, we will discuss recent joint work with Hyde and Schul that provides quantitative analogues of this statement.
 

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In the past two decades, starting with the pioneering work of Bourgain, Brezis, and Mironescu, there has been widespread interest in characterizing Sobolev and BV (bounded variation) functions by means of non-local functionals. In my recent work I have studied two such functionals: a BMO-type (bounded mean oscillation) functional, and a functional related to the fractional Sobolev seminorms. I will discuss some of my results concerning the limits of these functionals, the concept of Gamma-convergence, and also open problems.