Research

My doctoral dissertation dealt with the theory of maximal Cohen-Macaulay modules. In particular, I have been focused on determining if countable Cohen-Macaulay type implies finite Cohen-Macaulay type over a complete local Cohen-Macaulay ring with an isolated singularity. I am also interested in bounding projective dimension and various homological questions. Another area that I find very intriguing is the decomposition theory of Boij and Söderberg. I also enjoy working with Macaulay2. Below you will find a list of publications and maybe come current projects.

Publications

1. Calculations involving symbolic powers.
   The Journal of Software for Algebra and Geometry, Vol. 9 (2019), 71-80.
   Joint with B. Drabkin, E. Grifo, and A. Seceleanu.

2. Recursive strategy for decomposing Betti tables of complete intersections.
   International Journal of Algebra and Computation, Vol. 29, No. 7 (2019).
   Joint with C. Gibbons and R. Huben.
   [arXiv:1708.05440]

3. Advising undergraduate research in prison.
   Mathematical Outreach: Explorations in Social Justice Around the Globe, 255-263.
   Series on Mathematics Education: Vol 16, World Scientific, November 2019.

4. Visualizing combinatorial objects in Macaulay2.
   Sém. Lothar. Combin. 80B (2018), Art. 97, 6 pp.
   Joint with B. Barwick, T. Enkosky, and J. Vallandingham.

5. Generalized Multiplicative Indices of Polycyclic Aromatic Hydrocarbons and Benzeniod Systems.
   Zeitschrift für Naturforschung A, 72.6 (2017): 573-576.
   Joint with V.R. Kulli, Shaohui Wang, and Bing Wei.
   [arXiv:1705.01139]

6. Non-simplicial decompositions of Betti diagrams of complete intersections.
   J. Commut. Algebra 7 (2015), no. 2, 189-206.
   Joint with Courtney Gibbons, Jack Jeffries, Sarah Mayes, Claudiu Raicu, and Brian White. Results from a summer workshop in commutative algebra at MSRI, 2011.
   [arXiv:1301.3441]

7. A sequence defined by M-sequences.
   Discrete Math. 333 (2014), 35–38.
   Joint with Tom Enkosky.
   [arXiv:1308.4945]

8. Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type.
   Connections between algebra, combinatorics, and geometry, 299-317, Springer Proc. Math. Stat., 76, Springer, New York, 2014.
   [arXiv:1307.6206]

9. Super-stretched and graded countable Cohen-Macaulay type.
   Journal of Algebra 398C (2014), pp. 1-20.
   [Article], [arXiv:1301.3593]

10. Computing free bases for projective modules
    The Journal of Software for Algebra and Geometry (2013).
    Joint with Brett Barwick.
    [Article] , [M2]

11. Ideals with Larger Projective Dimension and Regularity.
    Journal of Symbolic Computation (2011).
    Joint with Jesse Beder, Jason McCullough, Luis Nunez, Alexandra Seceleanu, and Bart Snapp.
    [Article], [arXiv:1101.3368]

Macaulay 2 Packages

• Visualize.m2: Joint with Brett Barwick, Tom Enkosky, and Jim Vallandingham. This package helps visualize algebraic objects in the browser using javascript.

• Decompositions.m2: Joint with Courtney Gibbons during the MSRI Summer School in Commutative Algebra. This is a patch to supplement to the current Boij-Söderberg Macaulay2 package. This package allows the user to compute the coefficients of a Betti Table decomposition using the Herzog-Kohl equations and is included in the current version of the BoijSoederberg.m2 package in Macaulay2.

• QuillenSuslin.m2 This is joint with Brett Barwick. Given a projective module over a polynomial ring, this package uses Logar-Sturmfels’ algorithm to calculate the free basis.

• BigIdeal.m2 This package generates the ideals defined in Ideals with Larger Projective Dimension and Regularity by Beder, McCullough, Nunez, Seceleanu, Snapp and Stone. These ideals have very large projective dimension and regularity relative to the degree and number of generators.

Research with Undergrads

At Bard all students are required to write a senior thesis for each concentration they choose. This includes the incarcerated students in the Bard Prison Initiative. For privacy reasons, I am not willing to publish the BPI students work. However, if you are interested in veiwing it, feel free to contact me. The titles of the current students are tentative.

• Classifying nearly complete intersections Summer 2019

• Charlie Miller, Hamilton College

• Semi-definite programming with Macaulay2 Spring 2018

• Vincent Schinina, Adelphi University

• Linear programming with Macaulay2 Fall 2017

• Kyle Murray, Adelphi University

• Finding complex roots May 2017

• Nicholas DeMarco, Adelphi University (Co-advised with Sarah Wright)

• Walks on molecular graphs May 2017

• Marisa Masi, Adelphi University

• Matroids on Rings with applications to toric ideals May 2017
• Patrick Phelps, Adelphi University

• From String Theory to Elliptic Curves over Finite Field, $\mathbb{F}_p$ May 2014

• Linh Pham, Bard College

• Let’s Walk and Explore May 2014

• Bard College (BPI)

• A New Look at Hadwiger’s Conjecture May 2014

• Bard College (BPI)

• Concrete Bridges to Abstract Algebras May 2014

• Bard College (BPI)

• Sifting Squared Prime Intervals Efficient Prime Acquisition and Counting May 2014

• Bard College (BPI)

• Algebraic Structures and Boij-Söderberg Theory May 2013

• Fanny Wyrick-Flax, Bard College

• Applications of Graph Theory to Chaotic Systems January 2013
• Bard College (BPI)

• (Co-advised with Georgi Gospodinov)

• Computing Various Dimensions of Chaotic Systems January 2013
• Bard College (BPI)
• (Co-advised with Georgi Gospodinov)