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*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.*

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

In 1983, Edelsbrunner, Kirkpatrick and Seidel introduced the concept of the alpha shape, which is a piecewise-linear region surrounding a collection of points in the plane $\mathbb{R}^{2}$, including the 2-dimensional convex hull as a special case [3]. Edelsbrunner and Mucke extended the alpha shape to three dimensions, a construction which was later used to create beautiful models of the shapes of certain biomolecules.

I will share a new algorithm developed with J. Carlsson for computing a closely related combinatorial structure known as the alpha complex in higher dimension. The reason we began looking at the alpha complex has do with the discovery of a different type of shape associated to Gaussian mixtures, which is naturally approximated by weighted versions of the alpha complexes.

*For the full article, as well as example illustrations, check out our 2023 Department Newsletter! This article starts on page 6.*

This July, our Department hosted two conferences! Both were part of annual traditions, namely the Trisectors workshops and the Formal Power Series and Algebraic Combinatorics (FPSAC) annual meetings.

The Trisectors Workshop took place June 26-June 30 and was organized by Alex Zupan, Laura Starkston, Jeffrey Meier, Maggie Miller, and Gabe Islambouli. This year's workshop emphasized connections with symplectic topology. It was preceded by introductory lectures delivered over Zoom during the week leading up to the conference and featured several afternoon devoted to group projects.

The FPSAC meeting, a much larger event, took place July 17-21 and was organized by a much longer list of people including several from UC Davis: Monica Vazirani, Matt Silver, Anne Schilling, Dan Romik, Alex McDonough, Gladis Lopez, Fu Liu, Shelby Kustak, Sean Griffin, Tina Denena, Jesus DeLoera, and Eric Carlsson.

Both conferences were reported to be very successful! Let's thank our colleagues and staff for their hard work!

Learn more about these conferences at the websites for FPSAC 2023 and Trisectors Workshop 2023.

Moon Duchin and Dylan Thurston (erstwhile KAP and UCD faculty child respectively) brought the following question arising from gerrymandering:

Identify the subset $P_n$ of the simplex $\Delta$ in $\mathbb{R}^{n!}$ (viewed as all probability measures on permutations) obtained by fixing $n$ (typically different) probability measures on $\mathbb{R}$, sampling one point from each and recording the resulting permutation.

For example the midpoint or corners of $\Delta$ result if the measures are equal or each have distinct one point support respectively.

On the other hand $P_n$ is not all of $\Delta$. In particular a global quadratic inequality follows from the correlation between the conditions $a<b$ and $a<c$. Already for $n=3$ if $[abc]$ is the probability that $a<b<c$ then $[abc][cba]\leq([acb]+[cab])([bac]+[bca])$ so for instance the midpoint of the edge in $\Delta$ between $[abc]$ and $[cba]$ is not in $P_3$. Many such global quadratics have been written down by Fontain, Kasteleyn and Ginibre.

This ostensibly measure related problem becomes algebra upon noting that it suffices to consider measures of finite support and that for each fixed collection $\sigma=(\sigma_i)|_{i\in[n]}$ of disjoint finite support sets the associated region $P_\sigma\subseteq P_n$ is the image of the positive real points of a rational variety in $\mathbb{P}^{n!-1}$. For example with $n=3$ and $\sigma=(\{1,5\},\{2,4\},\{3\})$ there is a square of possible measures with these supports indexed by the probability $x$ for $1$ (rather than $5$) in the first measure and $y$ for $2$ in the second. The image $P_\sigma$ of this square in $\Delta$ is the ruled surface given by $[bac]=[cab]=0$ and $[abc][cab]=[acb][bca]$ and is the positive part of the Segre embedding of $\mathbb{P}^1\times\mathbb{P}^1$ into $\mathbb{P}^3$. More commonly the initial product of projective spaces will require some resolution away from the positive part before the rational map becomes algebraic.

A second connection to algebra is that when viewing $\sigma$ as a sequence of numbers from $[n]$ so that the example above becomes $abcba$ a braid move such as to $acbca$ does not change $P_\sigma$ though it does change the associated coordinatization. Thus it suffices to consider $\sigma$ indexed by elements of the $K_n$ Coxeter group. For $n=2$ this is affine type A and there are up to the action of $S_3$ only $\lceil\frac{k}{2}\rceil$ words of length $k$ required to get all of the sets $P_\sigma$ so for $k=5$ there are only $abcba$, $abcac$ and $abcab$.

Even for three measures there is more to the story. $P_3$ is full (five) dimensional in $\Delta$ and covered by the four $S_3$ orbits of $\{P_\sigma\}$ with $\sigma$ of length eight. The boundaries of these $P_\sigma$ besides the linear positivity requirements and the above FKG quadratics are defined by two other $S_3$ orbits of divisors (of degrees three and four) which we only found by computer and which do not give global inequalities for $P_3$. The analogs for higher $n$ remain quite mysterious.

The case of three measures is also essentially Efron's nontransitive or ro-cham-bo dice problem: Find three weighted dice (rock, paper and scissors) with various face values (these are the three measures) for which the probabilities that paper beats rock, scissors beats paper and rock beats scissors are all more than half. For example if you ask that all three of these probabilities agree and be as large as possible this will be a boundary point of $P_3$ along a diagonal line and contained in $P_{abcabca}$ which is a four dimensional divisor satisfying the quartic. (Since there are only two b's and c's the paper and scissors dice can be replaced with coins.)