91´ÎÔª

Shira Gilat – University of Pennsylvania

The Algebra of Supernatural Matrices

The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras. Supernatural matrices are a minimal solution to the equation of unital algebras Mn(X) ∼= X, which we compare to several similar conditions involving cancellation of matrices. This algebra has appeared under various names before, and it generalizes both McCrimmon's deep matrices algebra and m-petal Leavitt path algebra.

March 10, 2025 at 3:00 p.m. Math 103

Christian López Mercado – 91´ÎÔª, PhD Candidate

Reinventing Difficult Concepts in Advanced Mathematics

Many students struggle with abstract mathematical structures and concepts due to their highly formal nature and lack of intuitive entry points. In this talk, I explore how we can leverage the Realistic Mathematics Education (RME) framework to help students reinvent difficult concepts in number theory and abstract algebra. Through carefully designed instructional sequences, students work together to reinvent the concept of a group in abstract algebra and promote the learning of primitive roots in number theory, perhaps making these concepts more accessible and meaningful. Attendees will gain insights into practical strategies for implementing RME in advanced mathematics courses, ultimately bridging the gap between intuition and formal mathematical abstraction.

March 31, 2025 at 3:00 p.m. Math 103

Nhan Nguyen – Amazon Web Services

Prompt-Tuned Multi-Task Taxonomic Transformer (PTMTTaxoFormer)

Hierarchical Text Classification (HTC) is a subclass of multi-label classification, it is challenging because the hierarchy typically has a large number of diverse topics. Existing methods for HTC fall within two categories, local methods (a classifier for each level, node, or parent) or global methods (a single classifier for everything). Local methods are computationally expensive, whereas global methods often require complex explicit injection of the hierarchy, verbalizers, and/or prompt engineering. In this work, we propose Prompt Tuned Multi Task Taxonomic Transformer, a single classifier that uses a multi-task objective to predict one or more topics. The approach is capable of understanding the hierarchy during training without explicit injection, complex heads, verbalizers, or prompt engineering. PTMTTaxoFormer is a novel model architecture and training paradigm using differentiable prompts and labels that are learnt through backpropagation. PTMTTaxoFormer achieves state of the art results on several HTC benchmarks that span a range of topics consistently. Compared to most other HTC models, it has a simpler yet effective architecture, making it more production-friendly in terms of latency requirements (a factor of 2-5 lower latency). It is also robust and label-efficient, outperforming other models with 15\%-50\% less training data.

April 14, 2025 at 3:00 p.m. Math 103

Eric Chesebro – 91´ÎÔª

Kleinian groups: history and recent results

Kleinian groups, first studied by Felix Klein and Henri Poincaré in the late 19th century, are discrete groups of isometries of hyperbolic 3-space. Since then, they’ve become central objects in mathematics, with deep connections to fields including number theory, complex analysis, and geometric topology.

In this talk, I’ll share some of the historical development of Kleinian groups and aim to give a sense of what they are and why they matter. Along the way, I’ll highlight some examples and explain how these groups arise naturally in geometry and low-dimensional topology. I'll finish with more recent results and share connections to problems I worked on during my sabbatical in Australia and New Zealand.

April 28, 2025 at 3:00 p.m. Math 103

Elizabeth Gillaspy – 91´ÎÔª

Zappa-Szép products and Higher-Rank Graphs 

Zappa-Szép products are a generalization of a semidirect product, in which two groups act on each other compatibly to form a new group.  In recent years, the notion of Zappa-Szép product has been extended from groups to a wide variety of mathematical structures, including the higher-rank graphs which are the focus of my research.  In 2021, a summer research team consisting of UM students Adlin Abell-Ball, George Glidden-Handgis, and S. Joseph Lippert discovered a new Zappa-Szép like structure for higher-rank graphs, which does not quite coincide with the existing definition in the literature.  In this talk, I'll tell you about their work, and how it sheds new light on the structure of higher-rank graphs.

September 15, 2025 at 3:00 p.m. Math 103

Dan Finkel – Math for Love, Early Family Math

Anatomy of a Beautiful Math Experience

What makes a mathematical experience inspiring or empowering instead of tedious or miserable? In this presentation, we'll explore why people love and hate math, and how to create opportunities for meaningful mathematical learning. 

About the Speaker

Dan Finkel is a game creator, curriculum writer, puzzle author, and math evangelist. He gives talks nationally and internationally, and his TEDx Talk -  - has been viewed over a million times.  Dan’s Math for Love curriculum has been used by thousands of students, and is known for its combination of rigor and play. The math games he created with his wife, Katherine Cook, have won over 30 awards. They include Prime Climb, Tiny Polka Dot, Multiplication by Heart, and Rolly Poly. Dan is also the author of Pattern Breakers, a playful book of patterns for young children.  Dan is the Founder of , a Seattle-based organization devoted to transforming how math is taught and learned. Dan is also Cofounder of , which provides free, quality math resources for children from birth to 8.

October 15, 2025 at 3:00 p.m. Math 103

Faculty Evaluation Committee meeting

October 27, 2025 at 3:00 p.m. Math 103

Jason DeBlois – University of Pittsburgh

Things we know, and things we do not, about knots.

In this talk, by knots I will mean circles embedded in three-dimensional space. I’ll start with background and some history, spend some time on knot invariants, then turn to the geometric perspective on knot complements that was introduced by Thurston around 1980. Along the way, I’ll mention some significant open questions in the subject. I’ll finish by discussing recent progress on a couple of geometric questions, on hidden symmetries of and totally geodesic surfaces in hyperbolic knot complements.

November 3, 2025 at 3:00 p.m. Math 103

Javi Pérez-Álvaro – 91´ÎÔª

Matrix Methods in Data Analysis

I spent my sabbatical in Spain, in a town you’ve probably never heard of—Alcalá de Henares. Between walks, tapas, and too many coffees, I finished my book, Matrix Methods in Data Analysis. In this colloquium, I’ll share a bit about the town and a bit about the book.

November 10, 2025 at 3:00 p.m. Math 103

Junwei Liao – 91´ÎÔª

The Stern-Brocot Diagram and Linear Recursive Sequences

How can we visualize the convergence of rational approximations to any real number? We will use continued fractions, the iterative process that generates rational approximations, and the Stern-Brocot Diagram, which defines the unique structure and relationship between rationals in lowest terms. A Farey Recursive Function (\(\mathcal{F}\)) is defined on the points of the Stern-Brocot Diagram, where the \(\mathcal{F}\)-values of points lying on any line in the diagram satisfy a 2-term linear recurrence relation. For example, the Fibonacci sequence on the nonnegative integers can be extended across all rational numbers by a Farey Recursive Function. 

But what happens when we look at the \(\mathcal{F}\)-values of points that lie on a Euclidean line that is not a line in the diagram? 

This presentation introduces Cross Recursion, proving that sequences of \(\mathcal{F}\)-values along these Euclidean lines not in the diagram still follow linear recurrence relations. Our main result shows that the \(\mathcal{F}\)-values of points lying on a Euclidean line passing through \(\left(\frac{p}{q}, \frac{1}{q}\right)\) and \(\left(\frac{a}{b}, 0\right)\) is an interleaving of \(\varphi(w)\) linear recursive sequences, where \(\varphi(w)\) is the Euler totient function and \(w = |pb - qa|\). Each of these \(\varphi(w)\) sequences has an order at most \(w+1\). Furthermore, the \(\mathcal{F}\)-values of points lying on a horizontal line \(y=1/w\) is an interleaving of \(\varphi(w)\) linear recursive sequences, where each of these \(\varphi(w)\) sequences is a bi-infinite sequence with an order at most \(w+1\).

November 24, 2025 at 3:00 p.m. Math 103

Eric Marland – Appalachian State University & the 91´ÎÔª

The -ing of Mathematical Modeling

Teaching mathematical modeling is more than assigning problems based on open-ended, real-world scenarios or on reusing time worn techniques on slightly different problems.  It’s about guiding students through the process and helping them improve at each stage, from the earliest brainstorming steps all the way through to dissemination. Too often, only a single model is created with no interplay with the real world, leaving little opportunity for enhancement or refinement.  Too often, feedback comes only after the project is complete, leaving little opportunity for growth and reflection. How can we structure courses so students engage deeply with each step of the modeling process throughout the course? How do we provide rich problems that not only -can- be refined over several iterations, but provide the opportunity to actually work through those iterations?  This talk will offer an interactive look at these questions and more.

December 1, 2025 at 3:00 p.m. Math 103