# Research

My research is in commutative algebra, with a focus on its homological aspects. I have a particular interest in studying properties of commutative rings using theories of cohomological support and structures on resolutions. I often work with triangulated categories as they are well-suited to study such data, and it allows me to draw on ideas from a variety of other areas such as homotopy theory and representation theory.

Below are preprints submitted for publication:

Koszul homomorphisms and universal resolutions in local algebra with Ben Briggs, James Cameron, and Janina Letz (arXiv version)

High Frobenius pushforwards generate the bounded derived category with Matthew Ballard, Srikanth Iyengar, Patrick Lank, and Alapan Mukhopadhyay (arXiv version)

Below are my published articles:

Bounds on cohomological support varieties with Ben Briggs and Eloísa Grifo

Transactions of the American Mathematical Society (to appear): arXiv version

Resolutions of differential operators of low order for an isolated hypersurface singularity with Rachel Diethorn, Jack Jeffries, Claudia Miller, Nick Packauskas, Hamid Rahmati, and Sophia Vassiliadou

Michigan Mathematical Journal (to appear): arXiv version

Cohomological jump loci and duality in local algebra with Ben Briggs and Daniel McCormick

Mathematische Zeitschrift (2023): arXiv version, journal version

The homotopy Lie algebra of a Tor-independent tensor product with Luigi Ferraro, Mohsen Gheibi, Dave Jorgensen and Nicholas Packauskas

Illinois Journal of Mathematics (2023): arXiv version, journal versionA comparison of dg algebra resolutions with prime residual characteristic with Michael DeBellevue

Quarterly Journal of Mathematics (2023): arXiv version, journal versionA partial converse ghost lemma for the derived category of a commutative noetherian ring with Jian Liu

Proceedings of the American Mathematical Society (2023): arXiv version, journal version

Exceptional complete intersection maps of local rings with Srikanth Iyengar, Janina Letz, and Jian Liu

Pacific Journal of Mathematics (2022): arXiv version, journal version

Locally complete intersection maps and the proxy small property with Ben Briggs, Srikanth Iyengar, and Janina Letz

International Mathematics Research Notices IMRN (2022): arXiv version, journal versionConstructing non-proxy small test modules for the complete intersection property with Ben Briggs and Eloísa Grifo

Nagoya Mathematical Journal (2022): arXiv version, journal versionCohomological supports of tensor products of modules over commutative rings with Srikanth Iyengar and William Sanders

Research in the Mathematical Sciences (2022): arXiv version, journal versionSupport varieties over skew complete intersections via derived braided Hochschild cohomology with Luigi Ferraro and W. Frank Moore

Journal of Algebra (2022): arXiv version, journal versionCohomological supports over derived complete intersections and local rings

Mathematische Zeitschrift (2021): arXiv version, journal versionDuality and symmetry of complexity over complete intersections via exterior homology with Jian Liu

Proceedings of the American Mathematical Society (2021): arXiv version, journal versionEquivariant isomorphisms of Ext and Tor modules

Journal of Algebra (2020): arXiv version, journal versionThe derived category of a locally complete intersection ring

Advances in Mathematics (2019): arXiv version, journal version

Below is my PhD thesis: (Advisors: Lucho Avramov and Mark Walker)

Thesis (2019), Digital Commons PDF

Software:

I am a coauthor of the ThickSubcategories package for Macaulay2 with Eloísa Grifo and Janina Letz. (Note: This is still under construction but available for use, click the first link.)