Events of the Week

In this talk, we will survey the sheaf theoretic treatment of connections on smooth complex vector bundles. Here we explain the main tools to come out with a sensible definition of connection in full generality. Then we go on to see some of the properties of connections such as its local description and its transformation under a change of frame.

To be announced via the Math Seminars Listserv.

This talk presents an overview of ongoing research into representation theory over bands, which are combinatorial generalizations of fields, with applications in matroid theory and tropical geometry. A fundamental departure from the classical theory arises from the observation that many natural general linear objects in this setting are not invertible. To construct a general linear monoid, we study the multiplication of bimatroids and matrices over bands, investigate their interrelations, and construct their moduli spaces as combinatorial analogues of the determinantal variety. These constructions provide the foundation for defining band analogues of classical matrix groups and their representations. As a proof of concept, we conclude with a study of Schur functors in this new setting.

We begin the homology review: homology in the form the seminar needs it, developed from scratch, with no prior exposure assumed.
Chain complexes over a field, simplicial complexes and their homology, and a statement of the properties of singular homology we will use later. Time permitting, we reach contiguity of simplicial maps, the mechanism behind the stability of Rips persistence.

Solitons, or solitary waves, are well-known localised travelling wave solutions of integrable nonlinear PDEs with the special property that their intrinsic physical characteristics are unchanged when they interact with other solitons. It’s slightly less well-known that most integrable PDEs possess families of multi-soliton solutions. An idea that goes back to the earliest days of integrable systems is the concept of a soliton gas — an “infinite” collection of interacting solitons.
This idea has gained momentum recently to describe complex wave fields, particularly as tools have been developed to model a gas of solitons. In this talk, I’ll describe the history of soliton gases, some of the main questions and current methods for studying the problem, recent results, and, time permitting, some open questions.

Interpolation problems study the hypersurfaces in projective space that pass through a given number of points under certain constraints. Among the main questions are determining the minimum degree of such a hypersurface and understanding the number of independent conditions imposed by the points on hypersurfaces of a fixed degree.
In this talk, I will begin with the classical formulation of these interpolation problems and introduce some of the tools used to study them. We will then explore the setting where these points are assigned multiplicities and how this is encoded algebraically. This naturally leads us to the Demailly and Chudnovsky conjectures, which describe how the minimum degree grows asymptotically as the multiplicities grow. Finally, we will discuss the recent proofs of these conjectures and their connection to containment problems in commutative algebra.

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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.