Oded Schramm - Research

Research

A constant theme in Schramm's research was the exploration of relations between discrete models and their continuous scaling limits, which for a number of models turn out to be conformally invariant.

Schramm's most significant contribution is the invention of Schramm–Loewner evolution, a tool which has paved the way for mathematical proofs of conjectured scaling limit relations on models from statistical mechanics such as self-avoiding random walk and percolation. This technique has had a profound impact on the field. It has been recognized by many awards to Schramm and others, including a Fields Medal to Wendelin Werner, who was one of Schramm's principal collaborators, along with Gregory Lawler. The New York Times wrote in his obituary:

Schramm's doctorate was in complex analysis, but he made contributions in many other areas of pure mathematics, although self-taught in those areas. Frequently he would prove a result by himself before reading the literature to obtain an appropriate credit. Often his proof was original or more elegant than the original.

Besides conformally invariant planar processes and SLE, he made fundamental contributions to several topics:

• Circle packings and discrete conformal geometry.
• Embeddings of Gromov hyperbolic spaces.
• Percolation, uniform and minimal spanning trees and forests, harmonic functions on Cayley graphs of infinite finitely generated groups (especially non-amenable groups) and the hyperbolic plane.
• Limits of sequences of finite graphs.
• Noise sensitivity of Boolean functions, with applications to dynamical percolation.
• Random turn games (e.g. random turn hex) and the infinity Laplacian equation.
• Random permutations.