How to Build a Random Surface

Scott Sheffield, Massachusetts Institute of Technology and IAS
Fine Hall 314

In-Person and Online Talk

The theory of "random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics.   Just as Brownian motion is a special kind of random path, there is a similarly special kind of random surface, which is characterized by special symmetries, and which arises in many different contexts.

Random surfaces are often motivated by physics:  statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory.  But even without that motivation, one may be drawn to wonder what a "typical" two-dimensional manifold look likes, or how one can make sense of that question.

I will give an overview of what this theory is about, including many computer simulations and illustrations. In particular, I will discuss the so called Liouville quantum gravity surfaces, and explain how they are approximated by discrete random surfaces called random planar maps.