The Loewner Energy at the Crossroad of Random Conformal Geometry - Lecture 4

Explore the intricate connections between the Loewner energy, Schramm-Loewner evolution (SLE), and Weil-Petersson quasicircles in this comprehensive lecture. Delve into the mathematical concept of Loewner energy for Jordan curves, which originated from the large deviations of SLE, and discover how it measures the roundness of these curves. Learn about the finite nature of this energy in relation to Weil-Petersson quasicircles and its connection to determinants of Laplacians. Examine the role of Loewner energy as a Kahler potential on the Weil-Petersson Teichmueller space and investigate the more than 20 equivalent definitions of finite energy curves arising from various mathematical contexts. Gain insights into the links between Loewner energy, SLE, and Weil-Petersson quasicircles, while exploring their connections to other branches of mathematics. Understand how ideas from random conformal geometry inspire new findings on Weil-Petersson quasicircles and consider potential future research directions in this fascinating field.