Title: Optimal embeddings and extensions for Triebel-Lizorkin and Besov spaces in spaces in quasi-metric measure spaces.

Abstract: Embedding and extension theorems for certain classes of function spaces in $\mathbb{R}^n$ (such as Sobolev spaces) have played a fundamental role in the area of partial differential equations. In this talk, we will discuss some recent work which builds upon such results and identifies necessary and sufficient conditions guaranteeing that certain Sobolev-type inequalities and extension results hold for the scale of Triebel-Lizorkin spaces ($M^s_{p,q}$ spaces) and Besov spaces ($N^s_{p,q}$ spaces) in the general context of quasi-metric measure spaces. An interesting facet of this work is how the range of $s$ (the smoothness parameter) for which these inequalities and extension results hold is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. This talk is based on joint work with Dachun Yang and Wen Yuan. This talk will be accessible to graduate students having some basic knowledge of analysis.