I am a mathematician who specializes in homological algebra and commutative ring theory, with interests in the intersections of these subjects with algebraic geometry, homotopy theory, and noncommutative ring theory.
In my Ph.D. dissertation, I proved that local rings of minimal Hilbert-Samuel multiplicity may be identified through the structure of their Ext-algebras, a very sensitive homological invariant of a ring which contains all sorts of information. This established a link between the multiplicity of a local ring, which is of a distinctly geometric origin, and its Ext-algebra, which is a purely algebraic object. Ext-algebras that are Gorenstein or AS-regular were singled out in these results, two properties which are important in certain flavors of noncommutative projective algebraic geometry. The second part of my dissertation characterized local rings of minimal multiplicity in terms of a certain "homological linearity" property, which partially answered an open question in the theory of Koszul local rings.
After writing my dissertation, I moved on to a project which studied the Koszul complex and Koszul homology algebra of a standard graded commutative algebra over a field. I proved that certain types of homological linearity (related to the one mentioned in the previous paragraph) transfer in a very clean and controlled manner between a given standard graded algebra, its Koszul complex, and its Koszul homology algebra. This required the introduction of new definitions which capture these types of homological linearity, and I then proved that these linearity phenomena actually occur "in nature" by giving several examples.
My current projects are inspired by recent developments and trends in algebraic topology, and homotopy theory specifically, the former being the birthplace of homological algebra. One project in particular seeks to use homotopy-theoretic techniques to understand how the homological invariants of a given local ring vary as the local ring is deformed — to use a suggestive metaphor, I am asking: "Can the homological invariants of a ring be viewed as a 'continuous function' of the ring?" Other projects are inspired by operad theory and are attempting to place the homological linearity phenomena mentioned above in a conceptual framework that has already been used to explain the connection between the Sullivan and Quillen approaches to rational homotopy theory. One theme that runs through all of my current projects is an emphasis on tracking certain higher-order operations that have been known to exist for decades, but are becoming the focus of renewed interest due to recent advances in the theory of homotopy algebras and their applications.
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.
Linear resolutions over Koszul complex and Koszul homology algebras, J. Algebra 572 (2021), 163-194.
Items in preparation
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.