Characterizing Sobolev embeddings and extension results in metric-measure spaces

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, I will discuss some recent work which builds upon such results and identifies purely measure theoretic conditions that fully characterize certain embeddings and extension results for the scale of Sobolev spaces ($M^{1,p}$-spaces) in the general context of metric-measure spaces. The first part of this talk is based on joint work with Przemys\l{}aw G\'orka (Warsaw University of Technology) and Piotr Haj\l{}asz (University of Pittsburgh), and the results in the second part were obtained in collaboration with with Dachun Yang (Beijing Normal University) and Wen Yuan (Beijing Normal University). This talk will be accessible to graduate students having some basic knowledge of measure theory.