B.A. Mathematics (Writing minor), Malone University, 2010
M.S. Mathematics, University of Akron, 2013
Ph.D. Mathematics, University of North Carolina—Chapel Hill, 2019
Courses taught at H-SC
- Math 140: Calculus for Economics (Fall 2019, Spring 2020, Fall 2020)
- Math 243: Differential Equations (Fall 2019)
- Math 252: Transitions to Higher Mathematics (Spring 2020)
- Math 431: Algebraic Structures (Fall 2020)
Potential student project topics
See below for examples of what I mean by “combinatorial” and “algebraic” topics.
- Combinatorial topics:
- Mathematics of games
- Catalan numbers and Catalan objects
- Algebraic and combinatorial topics:
- Lie algebras and combinatorics
- Partially ordered sets (could be just combinatorial, or algebraic as well)
- I am also open to project ideas suggested by students!
A key to the terms used here, with some examples:
“Combinatorial”—combinatorics is a very broad area of mathematics dedicated to counting and describing structure. Here are two examples: (1) Counting: Suppose you are studying arrangements of playing cards (which would fall under the “Mathematics of games” topic). If you’re being dealt a 5-card hand, then there are 3,744 ways to be dealt a full house. This seems like a lot, until you realize that there are 2,598,960 possible 5-card hands, meaning that a full house is dealt only roughly 0.144% of the time, or 6 hands out of every 4,165 hands that are dealt. (2) Describing structure: Suppose you are studying the hierarchy of a company with 100 employees. If employee A directly supervises employee B, then put B below A on the page and draw a line between them. Do this for the direct supervisor relationships among all employees in the company, and you’ll wind up with the “Hasse diagram” that describes the partially ordered set that describes the internal structure of this company. Partially ordered sets can be used to describe these kinds of hierarchical structures, and their Hasse diagrams are great tools to visualize this structure. Prerequisites: None; just come with curiosity and a willingness to ask (and answer) good questions!
“Algebraic”—most relevantly, I mean linear algebra (matrices, linear transformations, vector spaces and bases, etc.). A student interested in this type of project should have some linear algebra first. Prerequisites: Strictly speaking, just linear algebra; it never hurts to have seen some other advanced mathematical topics as well. Projects in these areas would likely be a bit more advanced.
- Combinatorial Lie representation theory
- Minuscule representations of Kac—Moody algebras
- Partially ordered sets
Publications and preprints
Classifications of Γ-colored d-complete posets and upper P-minuscule Borel representations, arXiv: 2008.06745, submitted.
Unified characterizations of minuscule Kac—Moody representations built from colored posets, Electronic Journal of Combinatorics 27(2), #P2.42 (2020).
Recursive bijections for Catalan objects (with S. Forcey, M. Kafashan, and M. Maleki), Journal of Integer Sequences 16(5), (2013).