##### Overview

My research is in commutative algebra. I am mainly interested in maximal Cohen-Macaulay (MCM) modules over Cohen-Macaulay rings. In particular, my thesis is focused on studying matrix factorizations and MCM modules over hypersurface rings. If you are not familiar with the concept of a matrix factorization, here is a short explanation. They are an amazingly simple yet powerful tool originating in Eisenbud’s study of free resolutions of modules over hypersurface rings.

###### Links

##### Papers

###### Tensor products of d-fold matrix factorizations with Richie Sheng*

July 2024

Richie and I study the tensor product of d-fold matrix factorizations. Our main goal was to understand the decomposability of the tensor product operation. This work is motivated by Yoshino’s paper “Tensor product of matrix factorizations”. In the last section of the paper we show how to construct maximal Cohen-Macaulay and Ulrich modules over hypersurface domains. Constructing these modules is not new, though the tensor product gives a convenient way to work with them. What is new is being able to tell when these modules are indecomposable (or not) while also keeping track of their ranks, minimal number of generators, and multiplicity.

*Richie’s work on this project was funded through the Undergraduate Research Opportunities Program (UROP) at the University of Utah (summer 2024). He was also funded by the Utah Math Department (spring 2024).

###### Branched covers and matrix factorizations with Graham J. Leuschke

Bulletin of the London Mathematical Society (2023)

In this paper, Graham and I study the connection between the d-fold branched cover of a hypersurface ring and the category of matrix factorizations with d factors. We show that, from a representation theory perspective, these ideas are closely related. Namely, the category of MCM modules over the branched cover is representation finite if and only if the same is true of the category of matrix factorizations with d factors.

###### Matrix factorizations with more than two factors

February 2021

This paper is a study of the category of matrix factorizations with more than two factors. I start by describing a naturally occurring exact structure which turns out to be Frobenius. I determine the indecomposable projective-inejctive objects and I give explicit formulas for syzygies, cosyzygies, and cones.

The majority of the rest of the paper is focused on establishing two module theoretic descriptions of the category of matrix factorizations with d factors. This builds upon work of Knörrer and Solberg.

Several statements in this paper need updating (many hypotheses can be removed). My thesis has an improved organization of some of the results (for instance, the ring need not be complete to establish the Frobenius exact structure).

###### My Ph.D. thesis

May 2022

##### Software

###### ZZdFactorizations

with David Favero, Sasha Pevzner, and Keller VandeBogert

This is a package for working with d-fold (aka ZZ/dZZ-graded) matrix factorizations in Macaulay2. It will hopefully be available soon (fall 2024!?)

##### Talks

###### AMS Sectional Meeting - Special Session on Modules over Commutative Rings

Georgia Southern University, October 2024

###### Upcoming Researchers in Commutative Algebra (URiCA)

University of Nebraska, Lincoln, May 2024

My notes from the talk.

###### University of Utah Algebra Seminar

University of Utah, April 2024

###### Beijing Jiaotong University Algebra Seminar

Beijing Jiaotong University, February 2023 (virtual)

Ipad notes from the talk.

###### JMM - Special Session on Commutative Algebra

Seattle WA, April 2022

###### Commutative and Homological Market Presentations (CHAMP)

February 2022 (virtual)

Info about this talk can be found here. In particular, there’s a video on YouTube.

###### Route 81 Conference on Commutative Algebra and Algebraic Geometry

Cornell University, November 2021 (virtual)

I was invited to give a 30 minute talk at the Route 81 Conference hosted this year by Cornell. You can see my slides here if you are interested.

###### AMS Sectional Meeting - Homological Methods in Commutative Algebra

Creighton University, October 2021 (virtual)

I gave a 20 minute invited talk at the AMS special session on Homological Methods in Commutative Algebra in October, 2021 from these slides.

###### Syracuse Algebra Seminar

Syracuse University, October 2021

In October 2021 I gave an hour long talk at the Algebra Seminar at Syracuse. My notes are here.

###### UCGEN Seminar

April 2021 (virtual)

In April 2021 I was invited to give a talk at UCGEN, an online seminar focusing on Algebraic Geometry and related subjects. A video is on YouTube! I gave an hour long talk from these slides which can also be viewed below.

###### Syracuse MGO Colloquium

Syracuse University, September 2021

I gave an expository talk about the classification of simple hypersurface singularities and its connection to matrix factorizations. My (rough) notes are here.

###### Syracuse MGO Colloquium

Syracuse University, March 2021

In March 2021, I gave an expository talk about matrix factorizations for the graduate students in the math department at Syracuse. You can find my notes here.

###### BUGCAT

Binghamton University, 2020

I gave a short virtual talk about matrix factorizations at the Binghamton University Graduate Conference in Algebra and Topology. My notes can be downloaded here.