Title: Random Fractal Geometries

 

Abstract: Fractal objects appear ubiquitously in nature---across physics, geology, chemistry, biology, and beyond---and are often intertwined with randomness. In the first part of this talk, I will present examples of classical fractal curves and more intricate fractal geometries that arise in mathematics and the natural sciences. Many of these phenomena remain only partially understood from a rigorous mathematical standpoint.

 

In the second part, I will discuss a celebrated result of Lawler, Schramm, and Werner, who proved that the fractal dimension of the Brownian frontier (the outer boundary of planar Brownian motion) is 4/3, originally conjectured by Mandelbrot. I will outline the main ideas behind their proof, assuming undergraduate-level knowledge of complex analysis and probability.