Roulette with Curves and Surfaces

This lecture delves into the mathematical concepts behind building a roulette, exploring probability distributions on curves, and studying properties of random surface models. The instructor discusses statistical symmetries, the connection between Brownian motion and fractal curves, and the groundbreaking Schramm-Loewner evolution (SLE) processes. The lecture also covers the two-dimensional Gaussian freefield, its irregularity, contour lines, and geometric properties. Excursion decomposition of the Gaussian free field is explained, showcasing how it can be decomposed into zero sets and positive/negative sine excursions. The lecture concludes with an outlook on understanding random geometry in three dimensions and the mathematical intuition involved in exploring new concepts.