The Continuum Schaeffer BDG Bijection - 25 Years Later

Explore the continuous version of the Schaeffer and Bouttier-Di Francesco-Guitter bijections in this mathematical conference talk that examines how these powerful combinatorial tools extend to random metric spaces. Discover how canonical random metric spaces like the Brownian sphere can be constructed through continuous versions of classical bijections, with the Brownian sphere emerging as a deterministic function of the Brownian snake. Learn about groundbreaking research conducted with Omer Angel, Emmanuel Jacob, and Brett Kolesnik that addresses the fundamental question of whether a measurable inverse function exists - one that associates a continuous labeled tree to a measured metric space as the inverse of the "continuous bijection." Understand how the precise solution to this mathematical problem crucially involves the orientation of the Brownian sphere, representing a significant advancement in the field 25 years after the original bijection work. Gain insights into the fine properties of maps through bijective methods and their continuous extensions, bridging discrete combinatorial structures with continuous random geometry.