The Zoo of Discrete Conformal Structures

Explore the diverse landscape of discrete conformal structures in this Fields Institute lecture from the Workshop on the Geometry of Circle Packings. Delve into David Glickenstein's comprehensive analysis of Thurston's circle packing formulation and Luo's multiplicative edge weight approach. Examine key properties, convergence of discrete conformal mappings, and the formation of geometric structures on Poincaré duals. Investigate the classification of discrete conformal structures and their relationship to curvature variations and discrete Laplacians. Learn about applications to discrete conformal mapping problems and recent rigidity results. Gain insights into hyperbolic backgrounds, dual structures, and potential future directions in this field of geometric analysis.

Intro
Where am I?
FRG Workshop on Geometric Methods for Analyzing Discrete SE
Conformality from the Riemannian perspective
Conformal variation of curvature
Discrete geometric structure
Axioms for discrete conformal structure
Some (1) history of circle packing
Some history of multiplicative conformal structure
Inversive distance packings
Angle Variation of Discrete Conformal Structures
Proof of the variation formula (G. 2011)
Variational formulation
Classification of Discrete Conformal Structures
Proof of the classification
Local rigidity
Flexibility
Application to domains (G. 2016)
Discrete conformal structure in hyperbolic background
Hyperbolic dual structures (G. Thomas 2017)
Future directions