The Loewner energy for Jordan curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle. This class of curves has intriguingly more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and string theory, and has been studied since the eighties. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from probability theory inspire new results on Weil-Petersson quasicircles and discuss further directions.
Min-max minimal hypersurfaces with higher multiplicity
Recently, X. Zhou proved that the Almgren-Pitts min-max solution has multiplicity one for bumpy metrics (Multiplicity One Theorem). In this talk, we exhibit the first set of examples of non-bumpy metrics on the $(n+1)$-sphere ($2\leq n\leq 6$) in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$-sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. This is a joint work with X. Zhou.