Research

My scholarly interests lie in three main areas: the intersections of commutative algebra, algebraic geometry, and graph theory; games on graphs; and scholarly teaching, including the use of mastery assessments and inquiry-based learning.

Mathematics

My dissertation research was in the exploration of the so-called containment problem. In its most general form, one may ask, given a homogeneous ideal \(I\), for which \(m\) and \(r\) is the symbolic power \(I^{(m)}\) contained in the ordinary power \(I^r\)? In addition to exploring this question in my dissertation in the context of algebraic geometry, I explored a related question in the context of edge ideals \(I(G)\). In that case, certain invariants of \(I\) related to invariants of the underlying graph \(G\).

Alongside some undergraduate students, I’ve also explored games on graphs, particularly the Game of Cycles, first described by Francis Su in his book, Mathematics for Human Flourishing.

Note to students

If you are an undergraduate and interested in research, please reach out! While it’s probably best if you’ve had at least one proof-based mathematics course (such as Dordt’s Math 212), it’s not a requirement. Let me know you’re interested, and we’ll see what we can find.

Scholarly teaching

I am also very interested in the use of research-based methods for teaching and assessment. As I describe elsewhere, I am a proponent of active learning (thought of broadly) in all my courses, and I am also interested in the use of low-stakes, mastery-based assessments tied to clear standards for the purposes of encouraging mastery of fundamental skills and the development of a growth mindset in mathematics.

Publications

• Comparing Powers of Edge Ideals, J. Alg. Appl., Vol. 18, No. 10 (2019) (with T. Kamp and J. Vander Woude)
• Mastery-based testing in undergraduate mathematics, PRIMUS, Vol. 25, No. 5, 441-460 (2019). (with J.B. Collins, A. Harsy, J. Hart, K. Haymaker, A. Hoofnagle, J. Stewart Kelly, A. Mohr, J. OShaughnessy)
• Start a Math Teacher Circle: Connect K-12 Teachers with Engaging, Approachable, and Meaningful Mathematical Problems, The 21st Proceedings of the Association of Christians in the Mathematical Sciences. (with T. Clark, A. Harsy, D. Klanderman, M. Maxwell, S. Robbert)
• The Waldschmidt constant for squarefree monomial ideals, J. Algbr Comb, 2016, (with C. Bocci, S. Cooper, E. Guardo, B. Harbourne, U. Nagel, A. Seceleanu, A. Van Tuyl, and T. Vu)
• On the Fattening of Lines in \(\mathbf{P}^3\), Journal of Pure and Applied Algebra, 219(4):1055–1061, 2015,
• Containment problem for points on a reducible conic in \(\mathbf{P}^2\), Journal of Algebra, 394(0):120–138, 2013, (with A. Denkert)

Talks/Presentations

• On Symbolic Powers of Ideals, Conference on Unexpected and Asymptotic Properties of Algebraic Varieties, University of Nebraska-Lincoln, August 2023

• A first experience in a flipped Calculus II course, MAA Themed Contributed Paper Session on Flipped Classes: Implementation and Evaluation; Joint Mathematics Meetings; San Diego, CA, January 2018

• Cut Your Cake (And Eat It, Too!), MAA Themed Contributed Paper Session on ``My Favorite Math Circle Problem’’; MathFest 2017; Chicago, IL, July 2017

• An Overview of Specifications Grading, 21st ACMS Biennial Conference; Charleston Southern University, Charleston, SC, May 2017

• My Favorite Math Circle Problem: Cut Your Cake (And Eat It, Too!), 21st ACMS Biennial Conference; Charleston Southern University, Charleston, SC, May 2017

• Specifications Grading in a First Course in Abstract Algebra, MAA Themed Contributed Paper Session on ``Teaching Abstract Algebra: Topics and Techniques’’; Joint Mathematics Meetings 2017; Atlanta, GA, January 2017

• Improving proof-writing with reading guides, MAA Themed Contributed Paper Session on Encouraging Early Career Teaching Innovation; MathFest 2016; Columbus, OH, August 2016

• Implementing Specifications Grading in a Linear Algebra course, MAA Session on Assessing Student Learning: Alternative Approaches; Joint Mathematics Meetings 2016; Seattle, WA, January 2016

• The importance of \(\alpha\), MAA Invited Paper Session on Concrete Computations in Algebra and Algebraic Geometry; MathFest 2015; Washington, D.C., August 2015

• Symbolic Powers of Ideals: Problems and Progress (15 minutes), 20th ACMS Biennial Conference, Redeemer University College, Ancaster, Ontario, Canada, May 2015

• Symbolic Powers of Ideals: Problems and Progress (50 minutes), Mathematics Seminar; University of South Dakota, March 2015

• On the Fattening of Lines in \(\mathbf{P}^3\), Oberwolfach Mini-Workshop 1508a, Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems; Mathematisches Forschungsinstitut Oberwolfach; Oberwolfach, Germany, February 2015

• On an Application of Bezout’s Theorem, Colloquium; Calvin College, February 2014

• Ideals of almost collinear points in \(\mathbb{P}^2\), Special session on Interactions Between Algebraic Geometry and Commutative Algebra; Summer 2012 Meetings of the Canadian Mathematical Society; Regina, Saskatchewan, June 2012

• Containment problem for ideals of points on a reducible conic, Special session in Algebraic Geometry and Graded Commutative Algebra; Fall 2011 AMS Central sectional meeting; Lincoln, NE, October 2011

• Results on the containment problem of ideals of fat points, Algebraic Geometry Seminar; University of Nebraska-Lincoln, April 2010

• What is Algebraic Geometry? (50 minutes), Geometry and Physics on Graphs REU; Canisius College; Buffalo, NY, July 2009

• A Fifteen-Minute Survey of Nonunique Factorization, IMMERSE at the University of Nebraska-Lincoln, July 2008