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December 2023
Elena Fuchs on Apollonian Circle Packings

An excerpt from the 2023 Department Newsletter research article...

Seventeen years ago, at a birthday conference for Dorian Goldfeld in New York City, Jeff Lagarias approached Peter Sarnak after a talk he gave on the arithmetic of so-called thin groups. He told him that he had recently written a paper on the number theory of Apollonian circle packings together with Graham, Mallows, Wilks, and Yan, and that the group governing the symmetries of these packings (the Apollonian group) was a thin group and so, perhaps he might find it interesting. I was just starting as Sarnak’s Ph.D. student back then, and that conversation undoubtedly shaped the course of my career and served as a springboard to many deep and interesting works over more than a decade to come. When I met with him the following week, he told me he had just the project for me, and handed me the paper of Lagarias et. al., which came with a laundry list of open problems at the end. One of those open problems had to do with a potential local to global conjecture for Apollonian packings, and within a couple years of studying Apollonian packings I formulated and provided convincing data for a precise conjecture of this form together with my co-author Katherine Sanden, who was an undergraduate at the time. Everyone believed it, and several people, including myself, dreamed of proving it one day. Fast forward to this summer, when a paper of Haag, Kertzer, Rickards, and Stange showed that, in fact, the conjecture is false. The story is not over yet: while the conjecture is false as previously stated, there is still a version of it that is probably true and is still as hard to prove. It is riveting enough, in my opinion, to grace the pages of this newsletter.

For the full article, as well as example illustrations, check out our 2023 Department Newsletter! This article starts on page 8.