Vladislav Guskov: From Julia Sets to Brownian Motion: Setting the Stage for Matings

Abstract

In this pre-colloquium talk, we will lay the groundwork for the upcoming SMC colloquium “An Introduction to Matings” by exploring fundamental concepts across three directions: complex dynamics, hyperbolic geometry, and random geometry. First, we will look at polynomial dynamics, define Julia and Fatou sets, recall quotient spaces, and briefly discuss a Peano curve, an example of a space-filling curve. Second, we will review key definitions from hyperbolic geometry and recall Möbius transformations. Finally, we will consider the most canonical stochastic process—Brownian motion—and, time permitting, introduce its 2-dimensional analogue, the Brownian map.