Fredrik Viklund: Liouville quantum gravity and random geometry

How does one choose in a natural manner a two-dimensional surface at random? Motivated by problems arising in string theory, Polyakov proposed one way to do this in 1981. His proposal had a deep impact for instance on the development of conformal field theory and string theory in the ensuing years. But it was very far from rigorous and only recently have mathematicians succeeded in giving precise meaning to Polyakov’s ideas. The resulting Liouville Quantum Gravity random surfaces are much rougher than smooth surfaces, yet they can be analyzed and turn out to have remarkable (and perhaps unexpected) properties often involving links with Schramm-Loewner evolution curves. In the talk I will indicate some parts of these stories (assuming essentially no knowledge in probability), and give impressions of what kind of objects are studied in random conformal geometry.