# Dr Nathanael Berestycki

In the following table, contact information relevant to the page. The first column is for visual reference only. Data is in the right column.

Division: | Department of Pure Mathematics and Mathematical Statistics |

Organisation: | University of Cambridge |

Tags: | Fellowship: Early Career, Researcher, University of Cambridge |

Related theme: | Mathematical sciences Physical Sciences |

## Biography

I joined the University of Cambridge as a lecturer in 2007. I became a Reader in Probability in 2012. I am also a fellow of King's college. Prior to hat I was educated in France and did a joint PhD thesis at Cornell University (USA) and Ecole Normale Supérieure (Paris), and continued as a postdoctoral fellow at the University of British Columbia, Vancouver (Canada).

## My Fellowship

Many questions in 2D are thought to be well described by a certain universal scaling limit, Liouville quantum gravity. One of the key insights for the study of random surfaces is conformal invariance, a rich and unexpected symmetry. Though it started only recently, this fellowship is already invaluable to me as it allows me to completely immerse myself in this challenging topic.

The interface between probability, geometry and analysis is enjoying an intensely creative phase where new geometric aspects of random structures are discovered. The random structures considered are ultimately of interest as models in physics, or biology or computer science but are motivated at least as much by the possibility that they are fundamental, universal mathematical objects. A key example is the emerging theory of random planar geometry, which is the primary concern of this project. What does a typical, random surface look like? And what is, in fact, the 'right' notion of a random surface?

In this project, we aim to unify two distinct approaches to this question (discrete and continuous) and try to show that they form two sides of one same coin. Along the way, this would show that certain natural discrete notions of random surfaces gain a fundamental new set of symmetries (known as conformal invariance) in the scaling limit, i.e., when viewed from far away. This would identify a canonical model of random geometry, known as Liouville random geometry.

Liouville random geometry is believed to be fairly counter-intuitive. Its 'topological' and 'metric' dimensions do not coincide; in order to travel from A to B you must first travel towards C. Hence another main goal of this project is to develop new techniques to better understand the properties of this geometry.