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In an effort to encourage professional development for our faculty, the Dean's Office has created the Dean’s Faculty Fellows Program. The program honors outstanding achievements in early career faculty from across the college. The award was launched in 2020 and comes with a stipend to support research activities. One of this year's awardee's is our own Laura Starkston! Dean's Faculty Fellows give a public lecture during the second year of their fellowship, and this will surely an event worth attending. Congratulations Laura!

An excerpt from the 2024 Department Newsletter research article...

A tensor is a multi-way array of numbers. An order-1 tensor is just a vector u ∈ ℝⁿ. An order-2 tensor is a matrix M ∈ ℝ^{n1× n2} . An order-3 tensor is a 3-way array T ∈ ℝ^{n1× n2× n3} , and so on. [...]

Tensors have a bunch of applications in statistics and data science. For instance, certain datasets might naturally be represented as a 3-wayarray encoding 3-way interactions between 3 different variables. Another key example in statistics is the method of moments: given many samples of an n-dimensional random vector, it may be useful to compute the moments. The first moment is the mean (expected value), which is an n dimensional vector; the second moment is the n × n covariance matrix; the third moment is an n × n × n tensor; and so on. We are used to performing various primitive computations on matrices: eigenvalues and eigenvectors, singular value decomposition (SVD), low-rank approximation, and so on. For tensors of order 3 and above, the analogous operations tend to be much more difficult to compute, or even ill-defined.

For the full article, check out our 2024 Department Newsletter! This article starts on page 8.

An excerpt from the 2024 Department Newsletter research article...

Informally, a mechanical linkage is a system of rigid links (rods or bars) connected by ideal joints and moving in the plane or in space. This definition suffices for engineering purposes, and one can find it in some form in many engineering books. However, from the mathematical viewpoint, this is not a satisfactory definition. Mathematically speaking, an abstract linkage is a finite connected metric graph L = (G, l), a graph G together with a length function l which assigns to every edge e ∈ E(G) of G a positive real number, its length l(e). Given such a graph and a target metric space (X, d) (for the purpose of this article, X will be Euclidean space of some dimension), one de fines the realization space R(L, X), as the space of maps from the vertex set of G to the target space X, f : V(G) -> X, subject to the condition d( f(v), f(w) ) = l ([v, w]).

Here v, w ∈ V(G)are vertices of G and [v, w] ∈ E(G) are the edges connecting these vertices. In terms of the informal definition of a mechanical linkage, each realization f ∈ R(L, X) defines a system of rods f([v, w]) connected at joints f(v).

For the full article, check out our 2024 Department Newsletter! This article starts on page 5.

As many of us know, Steve Shkoller's life revolves around surfing and mathematics. To see how these two passions build on each other, take a look at Greg Watry's discerningly written feature in UC Davis' Letters and Science Magazine. “The best wave you can get as a surfer is what’s called getting barreled, where the wave pitches over you and the crest falls in this cylindrical shape. You’re inside that cylindrical tube and then the wave crashes and when that wave crashes, that’s a singularity.” — Shkoller

Mariel Vazquez has been elected as a Fellow of the American Association for the Advancement of Science (AAAS). She was selected based on her contributions in both outreach and research. Her research spans a broad area of mathematics including low-dimensional topology and knot theory with a view towards applications in molecular biology. She is part of the 150th cohort of inductees, and one of ten at UC Davis. Congratulations, Mariel!

Alex Wein has won a prestigious NSF CAREER award. He explains: "I'm very grateful to receive this funding, which will support two lines of research that I'm excited about. The first involves understanding fundamental tradeoffs between statistical resources (e.g. quantity of data) and computational resources (e.g. runtime) in large-scale statistical inference problems. The second involves finding optimal algorithms for various inference problems on tensor-valued data (e.g. an n-by-n-by-n array), including the 3-dimensional reconstruction problem that arises in cryo-electron microscopy."

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.

What sort of collections of circles fit into a larger circle without overlapping? This question has motivated mathematicians for millenia. Here at UC Davis, Elena Fuchs has studied Apollonian circle packings and found that the curvatures appearing in such families of circles exhibit certain combinatorial features. Her results have led to stronger conjectures, widely accepted among number theorists, but one of these conjectures turned out to be false. This was discovered (via extensive number crunching) during a summer research project involving two students (one graduate, one undergraduate) led by Katherine Stange at the University of Colorado in Boulder. An article in Quanta magazine tells the story of the students' discovery and provides mathematical details.

UC Davis' Letters and Science has selected Professor Roger Casals as a College of Letters and Science Dean's Faculty Fellow for 2023.

UC Davis' Letters and Science interviewed Professor Rishidev Chaudhuri about the details of his research, which spans neuroscience and math. The article, Perception Inception: Exploring Decision-Making in the Brain with Rishidev Chaudhuri, provides an accessible view into what he's researching, and why he finds it interesting.