What Is a Random Surface?

Explore the fascinating world of random surfaces in this comprehensive lecture by Scott Sheffield for the International Mathematical Union. Delve into the modern theory while examining its rich historical context, enhanced by computer illustrations and animations. Begin with a simple concept of gluing equilateral triangles and progress to complex mathematical constructs like the Brownian sphere, peanosphere, and pure Liouville quantum gravity sphere. Discover how these seemingly disparate concepts are interconnected, drawing from decades of mathematical and physics research. Investigate the intriguing notion of continuum random surfaces embedded in d-dimensional Euclidean space, and learn how this concept extends to higher genus surfaces, surfaces with boundaries, and decorated surfaces. Gain insights into the deep connections between classical graph theory, complex analysis, probability, representation theory, string theory, planar statistical physics, random matrix theory, and two-dimensional quantum gravity. Suitable for both newcomers and experts, this colloquium-level overview aims to provide a clear and accessible answer to the fundamental question: What is a random surface?