From the universality to the Kähler geometry of random curves

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 appearance of the random simple curve, called SLE, as the universal scaling limit of interfaces in conformally invariant 2D systems. On the other hand, a subfamily of relatively regular simple curves forms the Weil-Petersson Teichmueller 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.

In this lecture we will give an introductory overview of the link.

The second lecture on May 8, 13:15 - 14:45, Lipschitzsaal, Endenicher Allee 60, will focus on the applications and further development in exploring this link, in particular, the holography of the Loewner energy as a renormalized volume in hyperbolic 3-space.