Title: Smoothness of functions and the geometry of sets.

Abstract: The primary focus of this talk is on how the geometric makeup of a given ambient can directly influence the amount of analysis that the underlying space can support. As an illustration of the interplay between these two branches of mathematics, we will survey some recently obtained results pertaining to the properties of extension and embedding domains for the scale of Besov and Triebel-Lizorkin spaces ($N^s_{p,q}$ and $M^s_{p,q}$ spaces) in the general setting of quasi-metric spaces. In particular, we will provide examples of several environments, including fractal sets, which highlight how the range of the smoothness parameter, $s$, for which these embedding and extension results hold is intimately linked to the geometry of underlying quasi-metric space. This talk is based on joint work with Dachun Yang and Wen Yuan.