Mentoring

I've worked with many undergraduate and graduate (Master's) students on their theses and summer projects.  Below is a list of these students, together with a summary of their work.  I have also included links to the final product when possible, but unfortunately some have been lost to the sands of time!

Undergraduate Students

• James Sun (joint with John Holmes, 2023):  An analysis of the Black-Scholes-Merton formula for pricing options.  This is studied using both a stochastic PDE and a change of measure approach, and for several different option types.

• Steven Liao (URECA 2022): A comparison on the effectiveness of several different signature-based Gröbner basis algorithms.

• Rickey Huang (2022): A detailed study of La Scala and Stillman's paper on how to minimize Schreyer's resolution.

• Phoebe Yan (2022): A primarily expository project on the connection between Lyndon words, necklaces, and free Lie algebras, but motivated by a question related to acyclic closures of DG algebras.

• Roger Wang (2019): Investigated the concept of robust optimization in the context of linear programming; studied Soyster's method.

• Camille Wixon (2019): Studied the mathematical underpinnings of Brin and Page's original PageRank algorithm for ranking pages on the World Wide Web.

• Irina Viviano (2018): Investigated ways to visualize a function of a single complex variable, with special attention to polynomials.  Implemented a preliminary software system for doing this using OpenGL.

• Jack Garvey (2018): Investigated the economic origins of the Scarf complex of a monomial ideal.

• Junyi Xue (2017): Studied a connection between an operation performed on symmetric functions and the calculations of conserved quantities held by solutions to the Korteweg-De Vries equation.  This is a continuation of the project started below by Ryan Dougherty.

• Mike Annunziata (2016): Implemented a noncommutative version of Faugere's F4 algorithm for computing Gröbner bases.

• Ariel Hawley (2015): Investigated the Leontief Input-Output model in economics, its connection to the Perron-Frobenius theorem, and ran examples from data obtained from the Bureau of Labor Statistics, as well as Leontief's original paper in Scientific American.

• Steven Hemric (2014): Studied algebraic statistics as well as the Metropolis-Hastings algorithm in connection to strategies in the game Rock-Paper-Scissors-Lizard-Spock.

• Ryan Dougherty (2014): Studied a connection between an operation performed on symmetric functions and the calculations of conserved quantities held by solutions to the Korteweg-De Vries equation.  This project was also mentored by Brian Pigott, who was a postdoc at WFU at the time.

Graduate Students

• Bryant Collins (2025): Implementation of a C++ library for computing with path algebras and related objects.

• Jaxon Wheeler (2025): Constructing cubic AS-regular algebras of dimension three acted on by Pansera's family of Hopf algebras.

• J De Oliveira Pinheiro (2025) (joint with Grey Ballard in CS): Searching for fast matrix multiplication algorithms arising from tensor decompositions with positive rank symmetric component.  The search algorithm uses a combination of sparsification (via a Schur decomposition), rounding with tolerance, and the Gauss-Newton method.

• John Tolbert (2024) (joint with Hugh Howards and Thomas Kindred): Linear algebra problems arising from integer quadratic forms appearing in knot theory.

• Tommy Meek (2023): Haskell implementation of Buchberger's Algorithm.  Includes type-level definition of monomials, coefficient fields, etc.

• Yunmeng Wu (2023): Analysis and implementation of Anick-Green's resolution of simple modules over a path algebra in Macaulay2.

• Logan Gray (2021): Wrote a package in Macaulay2 to find canonical bases of resolutions according to the classification of grade three perfect ideals due to Avramov-Kustin-Miller and Weyman.

• Dakota White (2021): Added to a Macaulay2 package HopfAlgebras (by Colin Martin, mentioned below).  New features include faster algorithms over finite-dimensional Hopf Algebras, and computing the character table of a almost cocommutative Hopf Algebra (in the sense of Witherspoon).

• Desiree Martin (2020): Generalized the construction of the Taylor resolution of monomial ideals in k[x_1,...,x_n] to the case of a skew polynomial ring.  Also showed this new construction is a DG algebra, generalizing work of Gemeda.

• Colin Martin (2019): Wrote a Macaulay2 package HopfAlgebras that allows one to define Hopf algebras, have them act on noncommutative rings, and compute their rings of invariants.

• Cody Gilbert (2018): Explored the counterexample of Katthan (which disproved a "theorem" of Berglund-Jollenbeck) which showed that trivial Koszul homology algebra does not imply Golodness for monomial ideals.

• Dorian Lee (2018): An expository thesis on the topic of tensor categories.

• Mike Annunziata (2017): A continuation of his undergraduate thesis.  Continued to refine the noncommutative F4 algorithm developed there into a more full-fledged package, and added several new computations that were possible due to a generalization of a block ordering from the commutative case.

• Brad Hall (2017): Studied algebra generators and relations of the center of the ring of invariants of the action of S_n and A_n on the (-1)-skew polynomial ring in n variables over a field k.

• Terry Henderson (2016): An expository thesis on the history of the Cayley-Bacharach Theorem and sums of squares.

• Andrew Kobin (2015): A largely expository thesis on the Class Field Theory required to classify primes of the form x^2 + ny^2 for various n.  Also performed experiments in MAGMA to see for which n is it true that there are infinitely many times when both x^2 + ny^2 and y^2 + nx^2 are prime.  

• Justin Lyle (2015): Explored noncommutative complete isolated singularities, which are a generalization of the usual commutative case to quotients of skew-commutative power series rings.