Within the broader context of commutative algebra, I study infinite free resolutions of modules over graded rings. Polynomial rings over a field (and quotients thereof) are especially nice because they can be viewed as direct sums of finite dimensional vector spaces, and the same is true for their modules. Not only do such graded rings and modules arise in many settings, including the study of projective varieties, algebraic statistics, and combinatorics, but they also have beautiful properties that make their study interesting in its own right.

- Complete Modern Algebra (MATH 325W)
- Complete the following tutorials on the Home tab of web.macaulay2.com:
- Welcome Tutorial
- Basic Introduction to Macaulay2
- Mathematicians' Introduction to Macaulay2

- Read parts of Questions in Boij-Soederberg Theory by Daniel Erman and Steven V. Sam:
- Section 1 for background on the types of problems I work on
- Skim section 3 and subsection 9.3 for the particular type of results I'm currently interested in generating

- There are two foundational papers that drive my current research in this area. You need not read them in their entirety, but you may wish to get a flavor for the results by skimming through these papers:
- I have worked on these types of problems with other students, and you can see that work by viewing the following papers:
- arXiv:1507.08354 - Rational combinations of Betti diagrams of complete intersections (Willamette Valley Mathematics Consortium REU, 2015)
- (with Robert Huben '15)
- Classification of decompositions of Betti diagrams of codimension four complete intersections

- Download this list and complete the homework problems on the second page. Where helpful, specific references to (parts of) the above papers are included to help you focus your reading.

- Modules over short Gorenstein rings (with Luchezar Avramov and Roger Wiegand), in preparation
- NCAlgebra: A noncommutative algebra package for Macaulay2 (with Andrew Conner and Frank Moore), in preparation
- Classification of decompositions of Betti diagrams of codimension four complete intersections (with Robert Huben, Branden Stone, and Fanya Wyrick-Flax)
- A note on the Decomposition of Betti diagrams of complete intersections (with Robert Huben and Branden Stone), in preparation
- The Maximum Likelihood Degree of Toric Varieties (with Carlos Amendola, Nathan Bliss, Isaac Burke, Martin Helmer, Serkan Hosten, Evan D. Nash, Jose Israel Rodriguez, and Daniel Smolkin) - arXiv:1703.02251 preprint (submitted), Slides (for Section 7)
- Rational combinations of Betti diagrams of complete intersections (with Mike Annunziata, Cole Hawkins, and Alex Sutherland) - arXiv:1507.08354 preprint; - M2 code; To appear in Journal of Algebra and its Applications
- Grandma Makes Granola (with Dick Bedient);
*College Mathematics Journal* - Critical pebbling numbers of graphs (with Joshua Laison and Erick Paul)
*Journal of Combinatorial Mathematics and Combinatorial Computing*, Volume 99 (2016), 199â€“224, arXiv:1501.04236; - Non-simplicial decompositions of Betti diagrams of complete intersections (with Jack Jeffries, Sarah Mayes, Claudiu Raicu, Branden Stone, and Bryan White);
*Journal of Commutative Algebra*, Volume 7 (2015), no. 2, 189â€“206, arXiv:1301.3441 - The cone of Betti diagrams over a hypersurface ring of low embedding dimension (with Christine Berkesch,
Jesse Burke, and Daniel Erman) -
*Journal of Pure and Applied Algebra*, Volume 216, (2012), no. 10, 2256â€“2268, arXiv:1109.5198 - Fixing numbers of graphs and groups (with Joshua Laison);
*Electronic Journal of Combinatorics*, Volume 16 (2009), no. 1, Research Paper 39, 13 pp

- Mentor for the Willamette Mathematics Consortium REU, Summer 2015: Rings and Matrix Theory
- BoijSoederberg.m2, package reviser (with Branden Stone)
- NCAlgebra.m2, package author (with Andrew Conner and Frank Moore)