Featured News

Mathematics is all around us. In a recent article for UC Davis' Letters and Science Magazine, Greg Watry features the work of Roger Casals, illustrating observations on "The Mathematics of Daily Life." Inspired by a caustic created by light falling through a window, Casals contemplated singularities and the mathematical tools to describe and understand them. In Casals words, “Everything seems very smooth, but at some point, a transition happens ... The theory of singularities helps us understand these transitions.”

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."