I am a mathematician who specializes in homological algebra and commutative ring theory, with particular interests in the intersections of these subjects with algebraic topology and algebraic geometry.
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 focus is on a large multi-year project to extend the characteristic-zero theories of L-infinity algebras and coalgebras to fields of arbitrary characteristic. These latter objects are the homotopy-theoretic versions of Lie algebras and coalgebras, and they have applications ranging from commutative algebra, deformation theory, and homotopy theory, all the way to theoretical physics. A 100+ page monograph is in preparation which contains my extension of the theory, and a follow-up paper will contain applications to commutative algebra. This monograph will hopefully be made public by the end of 2021.
My planned future projects are mostly in the intersection of commutative ring theory and homotopical 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 seek to understand the larger ramifications of my extension of the theory of L-infinity (co)algebras mentioned above; for example, this latter work seems to imply that there is an extension of operad theory which would allow one to define homotopy algebras and coalgebras over certain monads and comonads. The details are yet to be worked out.
The overarching theme that runs through all of my research is the connectedness of mathematics. My favorite problems and theories are the ones that borrow techniques and inspiration from other fields that may, at first glance, seem completely unrelated.
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.
Items in preparation
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.