I'm the Don Tucker Postdoctoral Research Assistant Professor at the University of Utah working with the commutative algebra group. Before that I was a postdoc at the University of Edinburgh where I worked with Milena Hering. I completed my PhD at UC Berkeley under the direction of David Eisenbud. I like to think about algebra, geometry and combinatorics with a focus on minimal free resolutions and Betti numbers.

I am passionate about incorporating undergraduates into my research and love teaching courses at all levels. In 2008, I directed a summer program in Algebraic Geometry and in 2013 I led an REU on commutative algebra and combinatorics and have led one-on-one research projects with undergraduates in Utah. Please follow the links to teaching and research for information and for publications coming from these experiences.

Math 4800 (Introduction to Research "Representation Theory")

Math 3210 (Foundations of Real Analysis - Spring 2017)

Math 4030 (Foundations of Abstract Algebra - Fall 2016)

Math 2270 (Linear Algebra - Spring 2016)

Math 1260 (Multivariable Calculus - Fall 2015)

I usually base my course websites on Canvas, but copies of exams, syllabi, quizzes and worksheets are available in the **Course File Archive**

I enjoy working on research projects with undergraduates in a variety of settings. Please see Undergraduate Seminars below for the seminars I have run while at Utah. I have also led the following REUs:

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 Lyman, 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, Josh 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.
- Papers
- Presentations

Jimmy after his talk in the Utah Commutative Algebra Seminar

A graph whose toric ideal has primitive but not indispensible binomials (with 2013 REU students)

The Berkeley REU 2013

At the University of Utah I have organized undergraduate seminars each semester. In these weekly meetings, students took turns presenting material or solving exercises. The topics are listed below:

Noble exhibiting the fundamental group of Utah.

**(Spring 2017) Hyperbolic Geometry: ** Following lecture notes by Charles Walkden. Topics included - the upper half plane and disc models of hyperbolic space, Moebius transformations, geodesics, and the Gauss-Bonnet Theorem. This semester I invited Colin Adams (and his cousin-in-law Sir Randolph Bacon III) to visit the students and speak to the department.

** (Fall 2016) Knot Theory: ** Following The Knot Book by Colin Adams, students learned the basics of knots and links. The second half of the semester was devoted to open-ended projects where students read expository or research papers and gave presentations. Group project titles included "The Fundamental Group", "An Introduction to Lattice Knots and Lattice Stick Numbers", "Moebius Transformations, Geodesics, and Knot Energy", and "The part knot theory plays in quantum money."

**(Spring 2016) Groups and Combinatorics: ** A course going over the basic structures of groups via examples coming from geometry and number theory. We covered the dihedral group, Euclidean algorithm, and symmetric group. The final day culminated with a computation of the number of derangements in the symmetric group.

** (Fall 2015) Freshman Topology Seminar: ** (following First Concepts of Topology by Chinn and Steenrod.) The main goal was to introduce students to point set topology and proofs. This culminated with a proof of the intermediate value theorem.

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

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.

[13]. Lower bounds for Betti numbers of monomial ideals (with J. Seiner) * Submitted *

[12]. Koszul algebras defined by three relations (with H. Hassanzadeh, S. Iyengar)* to appear in a Springer INdAM Volume in
honor of Winfried Bruns. *

[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)* to Appear in Proc. Amer. Math Soc. *

[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) *

[12]. Koszul algebras defined by three relations (with H. Hassanzadeh, S. Iyengar)

[11]. The software package SpectralSequences (with N. Grieve, E. Grifo)

[10]. On the growth of deviations (with A. D'Alì, E. Grifo, J. Montaño, A. Sammartano)

[9]. Edge ideals and DG algebra resolutions (with A. D'Alì, E. Grifo, J. Montaño, A. Sammartano)

[8]. The closure of a linear space in a product of lines (with Federico Ardila)

[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)*

[6]. Robust toric ideals (with E. Robeva)

[5]. Free resolutions and sparse determinantal ideals

[4]. Formal fibers of unique factorization domains (with M. Daub, S. Loepp)

[3]. Dimensions of formal fibers of height one prime ideals (with M. Daub, R. Johnson, H. Lindo, S. Loepp, P. Woodard)

[2]. Sampling Lissajous and Fourier knots

[1]. On generators of bounded ratios of minors for totally positive matrices (with B. Froehle)

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

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.

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!