Research Interests

group theory, symmetric groups, graph labelings, magic configurations, Markov processes, Latin squares, machine learning, neural networks, artificial intelligence, representation theory, category theory, recreational mathematics, games, puzzles

My dissertation was on the Representation Theory of Hecke Algebras (which are closely related to Symmetric Groups). While I'm still interested in symmetric groups and representation theory, I've moved in a little bit more of a recreational direction more recently. I find projects in this area to be not only fascinating to think about (often requiring wide ranging mathematical concepts and tools), but they are also an excellent way to involve students in undergraduate research. A few of the articles below started as senior research projects that I advised. Afterwards, the student and I worked together to expand upon their earlier work.

In addition, I find that projects that involve puzzles or games are the best for explaining my research to non-mathematicians as well. While a mathematical layman probably won't understand the technical details of the proof, it is often very reasonable for them to understand the initial question(s) asked and the overarching answers. I must admit, it is quite satisfying to be able to demonstrate examples of real-world questions that mathematics can answer!

Mathematical Publications and Preprints

Each article below has either a direct link to a .pdf or a link to a location where the article can be viewed/downloaded.

  • Cross set: An Alternative Latin Squares Puzzle, in preparation (2020).
  • A New Take on Classic `Pen Problems', arXiv:1912.05646, submitted for publication (2019).
  • Markov Chains and Prime Climb, with Shaun Ceci, in preparation (2019).
  • Markov Chains and Trouble, with Shaun Ceci, in preparation (2019).
  • Non-magic Hypergraphs, with Benjamin Ellis, Jonathan Needleman, and Michael Raney, arXiv:1802.10392, submitted for publication (2017).
  • Minimal and maximal Numbrix puzzles, with Mary Grace Hanson, The Pi Mu Epsilon Journal, 14, No. 8, Spring 2018, 505-514 arXiv:1706.09389.
  • How do you fix an oval track puzzle?, with Sara Randall, Mathematics Magazine, 91 (2018) No. 5, 348-358 MathMag arXiv.
  • When are finite projective planes magic?, with Jonathan Needleman, Mathematics Magazine, 89 (2016) No. 2, 83-91. PDF
  • On Magic Finite Projective Space, with Jonathan Needleman, arXiv:1412.1545 Preprint (2014).
  • A basis theorem for the affine oriented Brauer category and its cyclotomic quotients, with Jonathan Brundan, Jonathan Comes, and Andrew Reynolds, Quantum Topology. 8 (2017), No. 1, 75-112. arXiv
  • Magic Fano Planes, with Benjamin Miesner, The Pi Mu Epsilon Journal, 14 (2014), No. 1, 23-29.
  • An interpretation of the Lascoux-Leclerc-Thibon algorithm and graded representation theory, with Alexander Kleshchev, Communications in Algebra, 38 (2010), No. 12, 4489-4500. PDF
  • Cayley Graphs of Symmetric Groups Generated by Reversals, The Pi Mu Epsilon Journal, 12:3, 143-147 (Fall 2005). PDF

Some Current Projects

Here are a couple topics that I'm currently looking into and/or preparing research papers on.

A sample level from the game Cross Set

Cross Set and Latin Squares

Cross Set is a puzzle game - with similarities to Sudoku - which was developed by Cuveet Story for the Steam gaming platform. In it, the player is given some partial information (which numbers could possibly appear in each cell of the grid) and is tasked with using logic to determine the unique Latin square which fits those clues. Drawing on the extensive work previously done on Latin squares, allows us to say nice thing about these puzzles as well.

A 4-regular hypergraph with a magic labeling over Z mod 4Z

Magic Hypergraphs and n3-Configurations

Just like magic squares, you can label the vertices on various types of hypergraphs or configurations and then call such a labeling magic when the sum along each line (or hyperedge) is constant. But why confine yourself to integers? This idea also makes sense when using the elements from any Abelian group as your labels.

Prime Climb Board

The Mathematics of Prime Climb

Prime Climb is a beautiful and fun mathematical game that was created through Kickstarter by Math for Love in 2014. On each turn you roll two 10-sided dice and apply the numbers one at a time to your current position(s) using the standard operations, +, −, ×, and ÷. The goal is to get your pawns to 101 and there are cool cards to draw along the way whenever you land on a prime larger than 10. This game begs all sorts of wonderful questions.