Bourgain Lecture

In probability theory, universality is the phenomenon where random processes converge to a common limit despite microscopic differences. For instance, the random walk, under mild conditions, converges to the same Brownian motion seen from afar, regardless of the law of each independent step. This phenomenon underlies the emergence of the random simple curve, called SLE, as the universal scaling limit of interfaces in conformally invariant 2D systems. SLE plays a central role in 2D random conformal geometry and a probabilistic approach to 2D quantum gravity and conformal field theory. On the other hand, a subfamily of relatively regular simple curves forms the Weil-Petersson Teichmüller space and has an essentially unique Kähler geometry. To describe these geometric structures we invoke the group structure and Kähler structure which is described via infinitesimal variations of the curves. Although these two worlds look very different we will explain how they are tied together via the Loewner energy.

I will give an introductory overview of the link, and discuss the applications and further development in exploring this link. In particular, we will mention the more recent result showing the relation between Loewner energy and the renormalized volume in hyperbolic 3-space (motivated by the AdS/CFT holographic principle), etc.