Home Teaching and Writing Research CV ☰

Research Interests

My research broadly concerns interactions between algebraic geometry, combinatorics, and commutative algebra. I am particularly interested in studying the way in which geometric information is preserved (or changed) upon deformation. Using these techniques I've been able to better understand the minimal free resolution and minimal generating sets of many classes of ideals arising from determinants, matroids, and graphs. It has also allowed us to better understand the way that the deviations of algebras (which are determined by its Poincaré series) behave.

Recently, I've been very interested in bounding the Betti numbers of certain classes of algebras. For instance in work with Srikanth Iyengar and Hamid Hassanzadeh we look at whether the upper bounds on the Betti numbers implied by the Taylor Complex hold for arbitrary Koszul algebras. On the other hand, with Jimmy Seiner, I've studied lower bounds for Betti numbers of monomial ideals.

Electronic copies of my papers are linked below.

Here is my CV.


In 2021 I gave a talk in the Fellowship of the Ring Seminar, the international virtual seminar for Commutative Algebra.

Links to Research papers

  1. On the Rank of Multigraded Differential Modules to appear in Mich. Math. J. (with J. Devries)
  2. Lower Bounds on Betti Numbers in Springer Volume in honor of David Eisenbud’s 75th Birthday (2021). (with E. Grifo)
  3. Techniques for predicting and avoiding unintentional biplanar movements during iliosacral screw placement Injury (2021) (with Ajinkya Rane, Bennet Butler, Robert O’Toole))
  4. Large lower bounds for the betti numbers of graded modules with low regularity (with D. Wigglesworth) Collectanea Math. (2021)
  5. Lower bounds for Betti numbers of monomial ideals (with J. Seiner) J. Algebra 2018
  6. Koszul algebras defined by three relations (with H. Hassanzadeh, S. Iyengar) Springer INdAM Volume in honor of Winfried Bruns 2017
  7. The software package SpectralSequences (with N. Grieve, E. Grifo) Submitted
  8. On the growth of deviations (with A. D'Alì, E. Grifo, J. Montaño, A. Sammartano) Proc. Amer. Math Soc.(2016)
  9. Edge ideals and DG algebra resolutions (with A. D'Alì, E. Grifo, J. Montaño, A. Sammartano) Le Matematiche (2015)
  10. The closure of a linear space in a product of lines (with Federico Ardila) J. Alg. Comb. (2016)
  11. Robust graph ideals (with B. Brown, T. Duff, L. Lyman, T. Murayama, A. Nesky, K. Schaefer) Ann. Comb. (2015)
  12. Robust toric ideals (with E. Robeva) J. Symbolic Computation (2015)
  13. Free resolutions and sparse determinantal ideals Math. Research Letters (2011)
  14. Formal fibers of unique factorization domains (with M. Daub, S. Loepp) Canad. J. Math (2010)
  15. Dimensions of formal fibers of height one prime ideals (with M. Daub, R. Johnson, H. Lindo, S. Loepp, P. Woodard) Comm. Algebra (2010)
  16. Sampling Lissajous and Fourier knots J. Experient. Math (2009)
  17. On generators of bounded ratios of minors for totally positive matrices (with B. Froehle) Linear Alg. Appl. (2008)

Undergraduate Research

I enjoy working on research projects with undergraduates in a variety of settings. If you are interested in an independent study course, please let me know! Below are some documents from previous work with students:
  • From 2018 - 2023, I have supervised many students on independent study reading course and research projects.
    • Noah Huang and Harrison Wolf (Summer Research Project on Pure Resolutions)
    • Brandon Eickert (Independent study on Algebraic Geometry)
    • Miranda Skinner (Independent study on Linear Algebra and Machine Learning)
    • Valerie Gilbert (Honors Thesis on Minimal Free Resolutions and Betti Numbers) pdf
    • Jonathan Cardenas (Independent study on Algebraic Geometry)
    • Noble Williamson (Honors Thesis, U. Utah) pdf
  • In Summer 2016, I supervised Jimmy Seiner (U. Michigan) on a project in commutative algebra and homological algebra. We continued working after the summer on variants of the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. Our paper is available here.
  • In 2013, I co-organized an REU at UC Berkeley with two other graduate students. We directed a total of 17 undergraduates in three research projects. The REU was funded by the Geometry and Topology RTG. My group of 6 students studied a problem in Combinatorial Commutative Algebra concerning toric ideals. Our paper Robust Graph Ideals appears in the Annals of Combinatorics. The students gave two presentations at a conference we organized with Stanford University. Their presentations:
    Presentation on Robust Graph Ideals
    Presentation on Regularity
    (Students advised: Bryan Brown, Timothy Duff, Laura Lymna, Takumi Murayama, Amy Nesky, Karl Schaefer)
  • In the Summer of 2008 I had the great joy to direct a reading REU at the University of Notre Dame. I worked with three undergraduate students (Joshn Mollner, Kaitlyn Moran and Emma Whitten) on the topic of algebraic geometry. The students learned a short course in algebraic geometry and wrote expository papers and presented at Mathfest. Their papers and presentations from MathFest are posted below. I'm really happy with how they turned out. I think they would be good for a first read in any of these interesting areas.