General Information

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Project Description

My project focuses on cataloguing and proving integrality and nonitegrality of the crystallographic circle packings that arise from Bianchi groups.

The paper detailing my research this summer, co-authored with my project partners Alisa Cui and Zachary Stier, can be found here.

For further information on our taxonomy of crystallographic packings, here is a link to our website.

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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.

Week 4:

We concentrated most of our energy on developing the database this week. I focused on building the Bianchi group section, specifically on generating the strip circle packings of each Bianchi group from every possible fruitful combination of their isolated clusters, on generating the bend matrices, and on uploading the Gram matrix and inversive coordinate data we generated last week. We also had a talk with Kei Nakamura exploring the geometric interpretation of Coxeter diagrams, and the translation of inversive coordinates into planes in hyperbolic space cutting out roots of reflection planes that define the space using Vinberg's algorithm. Alice Mark came to speak to us as well to explain the conceptual basis of Vinberg's algorithm as it relates to hyperbolic space.

Week 5:

I worked on producing strip diagrams and bend matrices for all the Bianchi groups, and getting all the information up on to the website. The code we've been using for strip diagrams is inefficient, and I began rewriting it, perhaps to be continued next week. The main task I dealt with this week and will continue to work on next week is producing bend matrices B that satisfy BV=VR, where V is a matrix containing the inversive coordinates of an isolated cluster (and part of its orbit, if necessary) and R is a reflection matrix for a specific circle in the cocluster (what we are reflecting the isolated cluster over) such that VR yields the inversive coordinates of reflected circles. V must be linearly independent, a step we had missed earlier in the week that I added into the code. We also learned about superintegrality of packings, which means that the bends (inverse of the radii) of circles produced by both the cocluster and the cluster acting on the cluster are integral. This is stronger than integrality, which means that the bends of circles produced only by the cocluster acting on the cluster are integral.

Week 6:

I began to work on proving integrality and nonintegrality for the Bianchi groups. Integrality and nonintegrality can be determined based on the inversive coordinates of the original cluster and cocluster of the packing, and a manipulation of the bend matrices (with a possible rescaling of the original inversive coordinates). I've also been reviewing the Belolipetsky-McLeod paper (mentioned above) to sharpen my understanding of the abstract concepts governing the Bianchi groups and their relationship to crystallographic circle packings.

Week 7:

I organized the Bianchi group packings into projected integral and nonintegral packings, and continued to work on proving integrality and nonitegrality. I ran into some bugs, but I hope to smooth them out and complete the proofs next week. I also prepared and presented the final presentation with my project coworkers, Alisa Cui and Zachary Stier.

Weeks 8 and 9:

My last two weeks were focused on co-authoring a final paper with my project partners, Alisa Cui and Zachary Stier, detailing our research this summer. A copy of the paper can be found here. Many thanks to NSF Grant DMS-1802119 and the Rutgers University-New Brunswick Math Department for funding, the DIMACS REU 2018 coordinators, and most of all Professor Alex Kontorovich, for such a fascinating summer.

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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