This is a page for information about my research and scholarship, including descriptions of several areas of work and links to various publications and other writings.

Mathematics of voting and democracy

In recent years, I have become very interested in the mathematics of voting methods, apportionment, and other topics related to democratic decision-making. This was spurred, in part, by beginning to teach a survey course on mathematics for humanities students at the liberal arts college where I work. I initially knew not much more than my students, but I have since read tons of books, articles, and blogs about mathematics, economics, social choice, history, politics, etc. and find it all fascinating. My main goal is to take this knowledge and reshape it for broader consumption. I’m intrigued by the mathematical nuances and pervasive paradoxes inherent to gathering informating and synthesizing it into a collective decision, but these are challenging ideas to grapple with, so I want to help the general public learn about and better understand these issues.

In September 2022, I published a book – “An Introduction to the Math of Voting Methods” – with 619 Wreath Publishing. I recommend it for use in a general education mathematics course or even a political science course with mathematically motivated students, but I also believe that anyone can pick up the book and learn from it. For another perspective on the book, check out this review from the UK’s Chalkdust Magazine, who selected it as a finalist for Book of the Year 2022.
I have also written a couple of blog posts for Insights, a blog on the Emmanuel College website. ** The first post – “The Mathematics of Ranked Choice Voting” – used a few concrete examples with small numbers to demonstrate how Ranked Choice Voting (“Instant Runoff”) works. ** The second post – “More on the mathematics of ranked choice voting and majoritarian winners” – elaborated on the first post and focused on a specific phenomenon with RCV: sometimes the eventual winner has a majority of continuing ballots, but not a majority of all ballots cast in the first round. In the post, I use a concrete example from Maine to illustrate how this could happen and why it’s not really a big deal (although some people may want you to think that it is…).

Numeracy and quantitative literacy

If you’re a mathematics educator who’s also interested in lesson plans that use current events, news articles, and social media posts to teach quantitative reasoning and mathematics content, check out the Eventmath project!

Eventmath is funded by a Wikimedia Project Grant.
Check out the site on Wikiversity and attend one of our future meetups to learn more about the project and get started on a lesson plan page of your own!
My wonderful collaborator Greg Stanton also published an article – “An Invitation to Eventmath” – in the Canadian Mathematical Society’s CMS Notes that describes the project goals and potential impact.
Please share these links with anyone else who might be interested in using current events and social to help students develop authentic, practical skills in mathematical reasoning!

Pursuit-evasion games on graph networks

Graph theory studies network connections. Think of a social web like Facebook, with nodes (called “vertices”) representing each person and connections (called “edges”) amongst the nodes representing who is friends with whom; in this sense, Facebook is a big graph.

“Cops & Robbers” is a game played on graphs. A team of Cops place themselves on the vertices of a given graph, and then the Robber places himself on a vertex. The two sides alternate turns: on their turn, the Cops get to move along the edges, then the Robber does the same, and they go back and forth like this. If a Cop lands on the Robber, he is caught and the Cops win. If the Robber is able to evade the Cops indefinitely, then he wins.

This game has been studied extensively since its introduction in the 1980s. Mathematicians have made great strides towards understanding what kinds of graphs allow one Cop to win, which graphs require more and how many are required, etc. However, there remain many open questions and unproven conjectures. This is a very active research area!

Since this is a game, this research can be purely recreational (as it is for me). But these results also have important applications and implications in computer science and programming. A good example is writing programs to search and organize large datasets efficiently. “Lazy Cops & Robbers” is a variant of the game wherein, on the Cops’ turn, only one of them is allowed to move. The main question becomes: Does this rule change the game significantly? For a given graph, do the same number of Cops suffice, or might we need more (and how many)?

For lots more information about this field of research, I highly recommend Anthony Bonato and his writings, especially his most recent book: “An Invitation to Pursuit-Evasion Games and Graph Theory.” Indeed, you can read my review of that book in the April 2023 issue of the American Mathematical Monthly.

And for even more information, please check out…

My Google Scholar profile
My ResearchGate profile
My faculty page on the Emmanuel College website
“Product Graphs and the Chaser-Runner Game”: YouTube recording of a 1-hour presentation on my research for the Talk Math With Your Friends seminar series (May 2021)
The Cops and Robbers Theorem: Infinite Series A YouTube video from PBS about this topic. I consulted on the script for the video.
Another video from PBS Infinite Series, a follow-up video to that first one.
The 3x3 rooks graph is the unique smallest graph with lazy cop number 3: https://arxiv.org/abs/1606.08485
An Introduction to Lazy Cops and Robbers on Graphs: https://www.tandfonline.com/doi/abs/10.4169/college.math.j.48.5.322

Other

I also have some expository presentations and writings, including the slides from my dissertation defense, posted on my old CMU personal page (which I, unfortunately, can no longer update).