(The Moebius function on the lattice of flats of matroid computes multi-graded Betti numbers of an associated ideal)

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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.

[14]. Large lower bounds for the betti numbers of graded modules with low regularity (with D. Wigglesworth) Submitted
[13]. Lower bounds for Betti numbers of monomial ideals (with J. Seiner) J. Algebra 2018
[12]. Koszul algebras defined by three relations (with H. Hassanzadeh, S. Iyengar) Springer INdAM Volume in honor of Winfried Bruns 2017
[11]. The software package SpectralSequences (with N. Grieve, E. Grifo) Submitted
[10]. 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)
[8]. The closure of a linear space in a product of lines (with Federico Ardila) J. Alg. Comb. (2016)

[7]. Robust graph ideals (with B. Brown, T. Duff, L. Lyman, T. Murayama, A. Nesky, K. Schaefer) Ann. Comb. (2015)
[6]. Robust toric ideals (with E. Robeva) J. Symbolic Computation (2015)
[5]. Free resolutions and sparse determinantal ideals Math. Research Letters (2011)
[4]. Formal fibers of unique factorization domains (with M. Daub, S. Loepp) Canad. J. Math (2010)
[3]. Dimensions of formal fibers of height one prime ideals (with M. Daub, R. Johnson, H. Lindo, S. Loepp, P. Woodard) Comm. Algebra (2010)
[2]. Sampling Lissajous and Fourier knots J. Experient. Math (2009)
[1]. On generators of bounded ratios of minors for totally positive matrices (with B. Froehle) Linear Alg. Appl. (2008)

Calculus Project

Two Calculus Projects that I have used in my Math 150 course. The first is a treasure-map oriented (mostly linear) project concerning our hero Percy Precal aand the second is an open-ended detective story about the mysterious orb incident in the land of Calculia

Commutative Algebra

Notes from the Undergraduate Program at Notre Dame in May 2019. The Lecture Notes and exercises offer an introduction to homological algebra and the Hilbert Syzygy Theorem. from a mini-course I ran at Notre Dame.

Number Theory Notes

- From Math 3 at the University of Edinburgh: Lecture Notes.

Algebraic Geometry

Some notes (ca 2007) on Algebraic Geometry from a mini-course I ran at Notre Dame.

• Lecture Notes - quick jog from the basics of ring theory to Hilbert Functions and Bezout's Theorem. (A next step might be this paper of Eisenbud, Green and Harris Cayley-Bacharach Theorems and Conjectures.)
• Notes on Free Resolutions

Workshop for High School Teachers

In Summer 2017, with Troy Jones and Ray Barton, I organized a workshop for high school teachers in Sandy, Utah. The theme was symmetry, and here is a link to some of the math activities we did each morning.

From Others

I have recently become a big fan of Mathologer on Youtube. My favorite video helps make sense of the "equality" $$ 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}$$ by first looking at different types of convergence and then discussing analytic continuations and the gamma function! For those interested in an introduction to higher level math without brushing anything under the rug - this is the channel for you!