Research Summary

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.

  • Metacognition in calculus: an effect on attitudes, w/Sarah Hanusch, to appear in the Proceedings of the 43rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Philadelphia, PA, October 14-17, 2021).

    • Abstract: Students often lack the cognitive and metacognitive strategies to maximize their learning. We redesigned a calculus course to teach students metacognitive strategies, with three components: frequent in-class discussions, corrections on exams, and a student essay on metacognitive strategies. We used a mixed method study design to qualitatively analyze the students’ essays and quantitatively measured changes in students’ attitudes towards mathematics using a pre-post assessment. We found the students attitudes improved at a practical and statistically significant level (p<0.0001) over the course.

Items in preparation

  • L-infinity (co)algebras in all characteristics: the homotopy transfer theorem, in preparation.

    • Abstract: The foundation of a theory of L-infinity (co)algebras over a field of arbitrary characteristic is laid, which closely parallels the well-established theory in characteristic zero. In particular, we offer a definition of L-infinity algebras which recovers the classical definition of differential graded Lie algebras (with squaring operations) when all higher brackets are trivial. The definitions are stated in terms of (co)brackets and higher Jacobi identities, and characterizations in terms of (co)differentials on appropriate (co)bar constructions are given. We define infinity-morphisms and describe their decompositions into arity components. Homotopy transfer theorems are proved, for both L-infinity algebras and L-infinity coalgebras, and formulas for the transfers are given in the differential graded cases. Applications to minimal models of L-infinity (co)algebras are given and the "inverse function theorems" for infinity-isomorphisms and infinity-quasi-isomorphisms are proved. Symmetric and divided power monads and comonads play a central role in the theory, along with cup multiplications and certain power operations in bigraded convolution algebras. A reconciliation with the characteristic-zero theory is described, and an extended introduction is included to serve as both an invitation to non-specialists and to help experts quickly identify the similarities and differences between our theory and the classical one.

  • L-infinity (co)algebras in all characteristics: universal envelopes and applications to commutative algebra, in preparation.

  • Deformations of homological invariants of local rings, in preparation.

Grant proposals

  • NSF RUI: Linking homotopical and commutative algebra through infinity-algebras, submitted.

    • Link to Project Summary here.