Degeneration of Singular Circle Patterns

Explore a comprehensive lecture on the degeneration of singular circle patterns presented by Xin Nie at ICBS2024. Delve into the intricacies of singular Euclidean circle patterns constructed from radius parameters assigned to triangulated surface vertices. Examine Thurston and Ge-Hua-Zhou's results demonstrating the cone-angle map's bijection from the parameter space to a convex polytope's interior. Discover the full description of the combinatorial structure of the "cone-angle polytope" and learn how to characterize degenerating sequences of circle patterns for every point on its boundary. Investigate the introduced concept of admissible partial edge orientation (APEO) and its one-to-one correspondence with polytope cells. Understand how each APEO defines a degeneration scheme for circle patterns and explore the generalization of Thurston's result, showing the bijection between the space of limiting objects from such degeneration and the cell corresponding to that APEO.