Research Summary
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. Lately, more and more of my attention has been aimed at understanding their role in the representation theory of hypersurface rings.
Links
Papers
Branched covers and matrix factorizations
with Graham J. Leuschke - October 2021
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.
Talks
JMM - Special Session on Commutative Algebra
Seattle WA, April 2022
Commutative and Homological Market Presentations (CHAMP)
February 2022 (virtual)
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
Syracuse University, October 2021
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.