# 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.

### About Maths Colloquium

The Mathematics Colloquium is directed at students and academics working in the fields of pure and applied mathematics, and statistics.

We aim to present expository lectures that appeal to our wide audience.

Information for speakers

#### Information for speakers

Maths colloquia are usually held on Mondays, from 2pm to 3pm, in various locations at St Lucia.

Presentations are 50 minutes, plus five minutes for questions and discussion.

Available facilities include:

- computer
- data projector
- chalkboard or whiteboard

To avoid technical difficulties on the day, please contact us in advance of your presentation to discuss your requirements.