Events of the Week

This talk explores the factorization of the polynomial \(x^m-1\) over finite fields, a classical problem that connects finite field theory, polynomial factorization, and field extensions. We begin with a brief introduction to finite fields and their basic properties, including the role of extension fields and primitive elements. We then introduce cyclotomic cosets and show how they organize the roots of \(x^m-1\), revealing the structure of roots of unity in finite field extensions. Using minimal polynomials of elements in finite field extensions, we derive an explicit factorization formula. The theory is illustrated through concrete examples, including the factorization of \(x^{13}-1\) over \(\mathbb F_3\) using a primitive element of the extension field \(\mathbb F_{27}\), and the contrasting factorizations of \(x^3-1\) over \(\mathbb F_2\), where it does not split completely, and over \(\mathbb F_4\), where it splits completely. We conclude by noting that the factorization of \(x^m-1\) has important applications in algebraic coding theory, particularly in the construction of cyclic codes—an area we will explore further in future presentations.

Consider a finite group acting linearly on a vector space. Each group element can be represented as a linear map whose graph is a subspace and the collection of these graphs gives a subspace arrangement. From an algebraic point of view, the vanishing ideal of the union of the subspaces can be used to find generators for the ring of polynomial invariants under the action of the group. Geometrically, we can also study the intersection lattice of the subspace arrangement, which can give cohomological information about the same ideal.
When one considers permutation actions, new combinatorial perspectives on this intersection lattice arise. In particular, for the regular action of the group on itself, the intersection lattice of the subspace arrangement is isomorphic to the coset poset, a known object of study in geometric and topological combinatorics. For other permutation actions, we establish that stabiliser subgroups of orbit partitions and their cosets can be used to label the vertices of the lattice. We then study these lattices with combinatorial techniques and computational tools to better understand the various algebraic structures associated with them.

Last week d_GH was defined as an
infimum of Hausdorff distances over all isometric embeddings of two compact metric spaces into a common ambient space: conceptually clean, but an infimum over a proper class. This week we prove the correspondence formula

d_GH(X,Y) = (1/2) inf_C dis(C), the infimum taken over correspondences C in X x Y. It removes the ambient space entirely and is the form in which d_GH enters every later estimate.
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The seminar is organized around one question; when are two metric spaces close, and which topological invariants respect that closeness? We take the Gromov-Hausdorff distance as the measure of similarity between metric spaces and work toward the stability theorem d_B(B(X), B(Y)) <= 2 d_GH(X, Y), which says that the persistence barcode is a Lipschitz invariant of the data.

Due to recent changes in federal regulation, all public-facing digital content, like what we post on our websites and the digital course materials we provide to our students, must be fully accessible to screen readers. This presents a particular challenge for us as mathematicians, since the traditional pdf output produced with Latex is generally not accessible.

In this talk, I will share what I learned on this matter in the past couple of months. The Good news is that the Latex community has found easy ways to help us. The Bad news is that some things are not quite doable yet. The Ugly news is that, unfortunately, some of our favorite Latex features might soon become a memory of the past.

We develop computational approximation theory on higher-genus domains, finite unions of disjoint subintervals of the real axis that arise naturally in orthogonal polynomial theory, iterative numerical linear algebra, and the study of integrable nonlinear wave equations. In each case, exploiting this underlying structure to choose appropriate basis functions (orthogonal polynomials) is essential to obtain a fast and accurate numerical method. A unifying theme is the numerical solution of Riemann--Hilbert problems on these domains.

It's a comparative study of various cryptographic protocols designed for 2-Party Secure Computation (2PSC) and, more precisely, for Private Set Intersection (PSI), with a particular focus on their implementation and performance under malicious adversary models. Among the protocols explored, the OPA-based and PSZ protocols were selected for detailed analysis due to their strong security guarantees and efficiency. Both protocols were implemented using Python and subjected to empirical testing. The OPA-based protocol employs algebraic constructions such as Oblivious Linear Function Evaluation (OLE), its enhanced version OLE+, and Oblivious Polynomial Addition (OPA), whereas the PSZ protocol leverages modern Oblivious Transfer (OT) extensions and Pseudo-Random Functions (PRFs). The study evaluates these protocols in terms of computational cost and communication overhead. The results offer a practical perspective on using such cryptographic primitives in real-world applications.

To be announced via the Math Seminars Listserv.

We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen--Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space $E_{3/2}^{+}(4N,\mathrm{id})$, a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula. This is joint work with Olivia Beckwith.

The distribution of special points in interesting geometric objects has been an active research topic. The Gauss circle problem and Linnik's problems are two well-known examples in this area. In this talk, we will discuss two particular results of this type and their variations. The first result, due to Duke, is about the distribution of CM points and geodesics in the upper half plane. The second one, due to Rudnick, is on the distribution of zeros of modular forms in the upper half plane.

Dynamic interface models arise in many physical and biological systems, including phase separation, multiphase flows, and cell and tissue dynamics. Their numerical simulation presents challenges in preserving underlying physical structures, reducing computational complexity, and accurately representing complex evolving geometries. In this talk, I will present structure-preserving numerical methods for phase-field models, structure-preserving reduced-order models for thermodynamically consistent PDEs, and geometric numerical methods for free-boundary problems, together with applications to biological systems.

To be announced via the Math Seminars Listserv.

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Many undergraduate students in non-STEM majors encounter mathematics primarily through procedural or remedial coursework that can reinforce negative beliefs about the subject. In this seminar we will examine the design of a puzzle-based general education mathematics course centered on mathematical sudoku, a genre that combines traditional sudoku with additional logical and mathematical constraints. Across three course units, students solve puzzles, document their reasoning, design and revise original puzzles, exchange feedback, and reflect on their mathematical experiences. We will consider how this structure is intended to support mathematical reasoning, creativity, productive struggle, and more positive relationships with mathematics. We will also discuss emerging results from a mixed-methods pilot study of two sections of the course taught in Fall 2025 at Clemson University. Paired pre-/post-survey responses yielded statistically significant increases in students’ beliefs that mathematics involves creativity and critical thinking, their confidence in their mathematical abilities, and their motivation to engage with mathematics. We will use these quantitative findings, together with qualitative results from reflective writings, to discuss the possibilities and limitations of puzzle-based general education mathematics, the measurement of attitudinal change, and implications for STEM education more broadly.