The fractional laplacian, extension problems and conformal geometry

The study of non-local operators has gained a lot of attention in the last decades due to their connections with probability theory and several applied models that take into account large range interactions. A particular example is the fractional laplacian in $\mathbb{R}^n$, a non-local operator that can be represented as  $$ (-\Delta)^s u(x) = c_{n,s} \int_{\rr^n} \frac{u(x+y)-u(x)}{|y|^{n+2s}} dy, $$ where $s\in(0,1)$ and  $c_{n,s}$ is a suitable constant. Note that, in contrast with differential operators, the value of $(-\Delta)^s u(x)$ is affected by the values of the function $u$ in regions that are far away from $x$. Recently, observed in the work of Chang and Gonzalez was a connection between the fractional laplacian and conformal geometry (and particularly with the AdS/CFT correspondence in string theory). In this talk I will describe some of the most relevant features of non- local operators, their connections with conformal geometry through extension problems and some further possible interpretations.