Student Projects

One of my favorite things to do at The College of Idaho is work with students on advanced undergraduate mathematics. You can check out some of the projects I have supervised below. I have several ideas for student projects that I’d be happy to explore with interested and motivated students.

Resources for My Students

Notes on Diagram Categories

Back in 2017 I wrote some notes to introduce undergraduates to diagram categories. The notes do not assume the reader has any experience with category theory, or even abstract algebra. The notes were intended to include a second part on Linear Diagram Categories, but that part was never realized. Perhaps some day I will go back and add this part, but the current version is still (I hope) a useful introduction to the topic.
notes

An Introduction to Nonstandard Look and Say Sequences

These notes were written to introduce students to the theory of look and say sequences in the spirit of John Conway’s paper The Weird and Wonderful Chemistry of Audioactive Decay. The notes are not meant as a replacement for Conway’s delightful explanation of his methods and results. Instead, they are meant to explain how Conway’s methods and results can be generalized to other (nonstandard) look and say sequences.
notes

The Python look_and_say module

I wrote a Python module that students (or anyone else) can use to explore nonstandard look and say sequences in the spirit of John Conway.
PyPI documentation GitHub Repo

My Students’ Projects

Jellyfish Brauer Categories related to special orthosymplectic supergroups

Ethan Bassingthwaite, Ryan Blau, Michael Coleman, and Julene Elias (2023-24)

In the Spring of 2023, this jellyfish crew studied the diagram category \(JB(2)\) which is equivalent to the full tensor subcategory of Rep(SO(2)) generated by the natural 2-dimensional representation of the special orthogonal group SO(2). The main result of the project was a diagram basis for \(JB(2)\) in terms of dotted Temperley-Lieb diagrams. They wrote up their result nicely in a paper. The following year, the four students continued their work, making good progress on similar theorems for jellyfish Brauer categories related to special orthosymplectic Lie superalgebras \(\mathfrak{sosp}(1|2)\) and \(\mathfrak{sosp}(2|2)\). In Spring 2024, the four students gave a poster presentation of all their results at the SRC.

paper

A Python Proof of Conway’s Cosmological Theorem

Ethan Bassingthwaite and Monika de los Rios (2022)

In the Spring of 2022, Ethan and Monika implemented a proof of John Conway’s Cosmological Theorem using Python. We followed the strategy of Zeilberger’s proof with implementation similar to that of Litherland. This proof was subsequently added to the Python module I wrote called look_and_say.
documentation

\(n\)th Root 2 Binary Look and Say Sequences

Santosh Acharya and Aspram Kharatyan (2021-22)

This project was completed during the Math Capstone course in the Fall of 2021. Santosh and Aspram explored various nonstandard look and say sequences coming from nonstandard binary number systems using base \(\sqrt[n]{2}\) for various positive integers \(n.\) Their results include descriptions of the growth rates and chemistry of the squences for particular seeds, as well as a Cosmological Lemma which can be used to completely describe the chemistry for all seeds at any particular choice of \(n\). Santosh and Aspram wrote up the details of their work in a paper and presented a poster summarizing their work at the 2022 SRC.
paper poster

Higher Dimensional Twindragons via Quaternions

Alex Tarasenko (2019 - 2020)

This project started with a reading course in the Fall of 2019 on Measure, Topology, and Fractal Geometry by Edgar. A particularly beautiful topic in that book is the connection between various representations of complex numbers and fractilings of the complex plane. The following spring, Alex explored higher dimensional analogs of this topic by replacing the 2-dimensional complex numbers with the 4-dimensional quaternion algebra (the Hamiltonians). Alex presented his results at the 2020 SRC. Unfortunately, I do not have a copy of Alex’s poster to share.

Factor groups of Knot and LOG groups

Samuel Chandler (2015 - 2017)

This project started as a summer reading on knot theory using The Knot Book by Adams. The following academic year, Sam learned to code in a computer algebra system called GAP in order to study factor groups of knot groups. Sam also studied quotients of generalizations of knot groups that arise from labeled oriented graphs. Sam presented his work at the 2016 SRC.
poster

Applications of Galois Theory to Differential Equations

Colton Grainger (2016)

Colton worked through the beautiful text Galois’ Dream: Group Theory and Differential Equations by Michio Kuga. This required Colton to apply methods from topology, algebra, and complex analysis in order to obtain a deeper understanding of the theory of differential equations. Colton presented examples at the 2016 SRC.
poster

Generators and relations for the Motzkin category

Jacob Karn (2016)

This project was an introduction to diagram categories. Jake worked through a draft of notes that I wrote, and ultimately proved a theorem which gives a nice presentation of the Motzkin category. Jake presented his work at the 2016 SRC.
poster