Published or to-appear articles
Preprints of my articles are available on the arXiv.
Homological criteria for minimal multiplicity, J. Algebra 523 (2019), 285-310.
Abstract: Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar.
Linear resolutions over Koszul complex and Koszul homology algebras, J. Algebra 572 (2021), 163-194.
Abstract: Let R be a standard graded commutative algebra over a field k, let K be its Koszul complex viewed as a differential graded k-algebra, and let H be the homology algebra of K. This paper studies the interplay between homological properties of the three algebras R, K, and H. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to K (and, more generally, to any connected differential graded k-algebra) and the other, called strand-Koszulness, which applies to H. The main theoretical result is a complete description of how these Koszul properties of R, K, and H are related to each other. This result shows that strand-Koszulness of H is stronger than Koszulness of R, and we include examples of classes of algebras which have Koszul homology algebras that are strand-Koszul.
Homotopical algebra through infinity-algebras. Notes from a talk delivered at the Syracuse University Algebra Seminar, October 15, 2021.
Three algebras and three definitions of Koszulness. Slides from a talk delivered at the "Route 81 Conference" at Queen's University, October 5, 2019, and "A Zoom Special Session on DG Methods in Commutative Algebra and Representation Theory," May 2, 2020. Based on the paper "Linear resolutions over Koszul complexes and Koszul homology algebras."
Curve singularities. For undergraduates. Explores the beautiful visuals that occur in the study of curve singularities. This is my attempt to write a math talk with as few symbols as possible.
Characterizations of minimal multiplicity via Ext-algebras. Slides from my Ph.D. thesis defense.
Ring theory & infinity. For undergraduates. An introduction to free resolutions in commutative algebra.