Research Overview

I am currently an Associate Professor of mathematics at High Point University. I was a math Ph.D. student at the University of North Carolina at Chapel Hill working with Professor Dima Arinkin. My Master's work focused on the formal reduction of differential operators to a canonical form. The paper is based on the classical papers by Turritin and Levelt, put into more modern language as given in books by Malgrange (as well as van der Put/Singer). A copy of the paper can be found here. My doctoral work was completed in May 2011, and my paper "Explicit calculations of local formal Fourier transforms" was accepted for publication in the journal Arkiv for Matematik, and can be found online here. The paper corroborates previous independent results by Sabbah and Fang, but proves them in a different way. A related article, "Explicit calculations of the local formal Mellin transform", was published in the Pacific Journal of Mathematics. A copy can be found here. 


My more recent research has been related to voting theory, gerrymandering and recreational mathematics. Specifically, I am interested in the prevalence of voting anomalies (such as instances of monotonicity or no-show paradoxes) in real-world ranked-choice voting elections, particularly those in the Bay area in California. My first publication in this area can be found here. I am also working with a group of researchers out of Duke University who are using outlier analysis to identify gerrymandering. We are currently investigating whether or not maps created by the Arizona independent commission of 2011 were skewed in any way. Lastly, I dabble in a number of recreational math topics such as the structure of the different end-behaviors found in the game Enemy Protector and strategies for winning the linear algebra game MatRixToe. I also wrote The Surname Impossibility Theorem which demonstrates the mathematical intractability of choosing the perfect surname for one's child. For a more detailed summary of my mathematics interests, take a look at my Research Statement.

Undergraduate Research

2021 (Summer): Vicky Sanderson and I are investigating voting anomalies related to Instant-Runoff Voting. Specifically, we are looking into generalizations of necessary and sufficient criteria for Monotonicity and No-show anomalies for 3- and 4-candidate elections, with ballots that are fully-ranked and not.
2019-2020: Camarie Schmidt is looking into the optimal outlier to drop in order to most improve a set of scores (for example, a set of homework assignments). While trivial if all assignments are scored equally, the question is much less obvious when assignments are weighted differently.
2019 (Spring): Sophie Kestner and I investigated how to apply efficiency gap calculations (used to measure gerrymandering) to situations with more than two parties. Sophie presented at High-PURCS.
2018-2019: Matt Knipfer and I worked on modeling ranked-choice elections and analyzing a new voting method. The big challenge of studying real-world ranked-choice elections is that there is so little data available to analyze. The way researchers get around this is to create models for the data and then analyze those models (for, say, the prevalence of monotonicity anomalies). Matt and I worked on a model to accurately match the truncated data typically produced in real-world IRV elections. We also investigated a voting method blending elements of Borda count, instant-runoff, and approval voting, to see which fairness criteria it would satisfy and fail. Matt also presented at MAA-SE.
2017-2018: I worked with Joanna Fass to investigate optimal strategies for different players in the game MatRixToe, over a variety of different rule structures. Joanna presented her research at MAA-SE and won a best-presentation award. Which I probably shouldn't take credit for, but I will anyway.
2016: I worked with Nick Zayatz and David Naylor on the prevalence of monotonicity anomalies in real-world instant-runoff elections. They created a program that looked for monotonicity anomalies in data from San Francisco and Alameda county (in California), and Burlington, Vermont. I have since built on that program to run elections using other voting methods and looked for other kinds of anomalies such as the no-show paradox. David and Nick presented on their findings at multiple conferences, and have gone on to fulfilling jobs in computer science which have nothing to do with voting theory.

SoTL Research

I have been involved in numerous aspects of the scholarship of teaching and learning, including writing projects and flipping the classroom. A paper I co-wrote with Elin Farnell and Julianna Connelly Stockton, Mat-Rix-Toe: Improving writing through a game-based project in linear algebra has been published in the journal PRIMUS. Another paper (with Laurie Zack, Jenny Fuselier, Ron Lamb, and Karen O'Hara) titled Flipping Freshman Mathematics was also published in PRIMUS. A third paper, related to the impact of mathematical experience of a student's cohort in a class, is also in PRIMUS. I really like PRIMUS, in case you couldn't tell. That paper is Environmental Impacts: how comparative prior knowledge affects students’ Calculus experience, co-authored with Lisa Carnell, Karen O'Hara, and Lindsay Piechnik. 


I also worked on a SoTL project with Jenny Fuselier investigating the benefits of a metacognitive intervention on student improvement in introductory math courses. We gave a presentation about metacognitive study habits (along the lines of prior work done by Saundra MacGuire in Chemistry classes) to students who had done poorly on their first test in a math class. We then followed the students to see whether or not the presentation helped them improve their test scores over the course of the semester.


Publications


--(with N. Zayatz) Lack of voting anomalies in empirical data of instant-runoff elections , Representation. DOI: 10.1080/00344893.2020.1785536
--The Surname Impossibility Theorem, Journal of Humanistic Mathematics, Volume 10 Issue 2 (July 2020), pages 222-236. Available at: https://scholarship.claremont.edu/jhm/vol10/iss2/11/.
--(with L. Carnell, K. O’Hara, and L.C. Piechnik) Environmental Impacts: how comparative prior knowledge affects students’ Calculus experience, PRIMUS (2019). DOI: 10.1080/10511970.2018.1472682
--Explicit calculation of local formal Mellin transforms, Pacific Journal of Mathematics (2016). DOI: 10.2140/pjm.2016.283.115
--(with L. Zack, J. Fuselier, R. Lamb, and K. O'Hara) Flipping Freshman Mathematics, PRIMUS (2015). DOI: 10.1080/10511970.2015.1031302
--(with E. Farnell and J. Stockton) Mat-Rix-Toe: Improving writing through a game-based linear algebra project, PRIMUS (2014). DOI: 10.1080/10511970.2013.876476
--Calculation of local formal Fourier transforms, Arkiv for Matematik (2013). DOI: 10.1007/s11512-011-0156-2