How Round is a Jordan Curve?

The Loewner energy for simple closed 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 intriguing class of curves has 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 math it touches on. I will also highlight how ideas from probability theory inspire new results on Weil-Petersson quasicircles and discuss further directions.