Keenan Crane, Carnegie Mellon University

Angle-preserving or conformal maps are a cen- tral tool used throughout analysis and geometry. Recent efforts to establish a faithful discrete analogue of conformal maps, beginning with Thurston’s circle packing conjecture, have recently culminated with a notion of discrete conformal equivalence that faithfully preserves much of the structure found in the smooth setting and comes with a complete uniformization theorem for discrete surfaces. This first lecture will provide an overview of the various perspectives on discrete conformal maps, leading up to discrete uniformization. Along the way, we will encounter some fascinating connections between combinatorics and geometry, as well as variational principles linking discrete conformal maps to realization problems for ideal hyperbolic polyhedra