General Information

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Project Description

My project focuses on the relationship between Bianchi groups under Mobius transformations and the Apollonian circle packing.

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Weekly Log

Week 1:

This week I met with my mentor, Professor Alex Kontorovich, to learn the background information necessary to begin my project, covering topics such as circle inversions, Coxeter diagrams, and Descarte's Kissing Theorem. I will be working on the relationship between Bianchi groups under Mobius transformations and Apollonian circle packings. I have been reading this paper by Katherine E. Stange on the topic.

Week 2:

Professor Kontorovich continued to prepare my project group for our research. We focused on the realization of polyhedra and their duals as circle packings, whose radii can be uniquely determined through minimizing the Bobenko-Springborn energy functional. The process is outlined in this paper by Gunter M. Ziegler. We then wrote a program in Mathematica that takes the faces and the number of vertices as parameters and outputs a diagram of the circle packing and the associated Gram matrix.

Week 3:

We learned about bend matrices, and began our exploration of Bianchi groups represented as quadratic forms and transformed into circle packings via Vinberg's algorithm. We're using McLeod's thesis collection of Vinberg algorithm inversive coordinates (with some error corrections) to concretely translate these forms into circle packings. This paper by Beloliptesky and McLeod has also been helpful in building circle packings from Bianchi groups via Vinberg's algorithm. We began building an online database cataloguing polyhedra, Bianchi groups, and higher dimensional packings with their Gram matrices, inversive coordinates, and supporting diagrams.

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Presentations

• First Presentation
• Second Presentation

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Additional Information

• My Mentor
  • Professor Alex Kontorovich
• My Coworkers
  • Alisa Cui
  • Gabriel Eiseman
  • Zachary Stier