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. I am interested in continuing this research program with an emphasis on understanding ring theoretic properties in terms of cohomological support and triangulated categories, as well as other homological invariants.
Below are my published articles (listed by journal publication date):
The derived category of a locally complete intersection ring
Advances in Mathematics (2019): arXiv version, journal versionEquivariant isomorphisms of Ext and Tor modules
Journal of Algebra (2020): 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 versionCohomological supports over derived complete intersections and local rings
Mathematische Zeitschrift (2021): 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 of tensor products of modules over commutative rings with Srikanth Iyengar and William Sanders
Research in the Mathematical Sciences (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 versionLocally 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 versionExceptional complete intersection maps of local rings with Srikanth Iyengar, Janina Letz, and Jian Liu
Pacific Journal of Mathematics (2022): arXiv version, journal version
A 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
A comparison of dg algebra resolutions with prime residual characteristic with Michael DeBellevue
Quarterly Journal of Mathematics (2023) arXiv version, journal versionThe homotopy Lie algebra of a Tor-independent tensor product with Luigi Ferraro, Mohsen Gheibi, Dave Jorgensen and Nicholas Packauskas
Illinois Journal of Mathematics (to appear) arXiv version
Below are preprints submitted for publication:
Cohomological jump loci and duality in local algebra with Ben Briggs and Daniel McCormick
arXiv versionResolutions of differential operators of low order for an isolated hypersurface singularity joint with Rachel Diethorn, Jack Jeffries, Claudia Miller, Nick Packauskas, Hamid Rahmati, and Sophia Vassiliadou
arXiv versionBounds on cohomological support varieties with Ben Briggs and Eloísa Grifo
arXiv version
Below is interesting work in progress:
High Frobenius pushforwards generate the bounded derived category joint with Matthew Ballard, Srikanth Iyengar, Patrick Lank, and Alapan Mukhopadhyay
Below is my PhD thesis:
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.)