I am interested in the use of free resolutions in algebraic geometry and commutative algebra,
and in the moduli of finite rank algebras.
In particular, I am interested in the structure of minimal free resolutions and in the use
of syzygetic techniques to study moduli spaces such as the Hilbert scheme of points
or moduli spaces of finite covers.
I completed my PhD in May 2010 at the University of California, Berkeley, under the supervision of David Eisenbud.
My NSF sponsor is Ravi Vakil.
Shapes of free resolutions over a local ring,
joint w/ Christine Berkesch, Manoj Kummini, and Steven V Sam.
( arXiv: 1105.2244.)
We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal \QQ-Betti sequences.
Tensor complexes: Multilinear free resolutions constructed from higher tensors,
joint w/ Christine Berkesch, Manoj Kummini, and Steven V Sam.
( arXiv: 1101.4604.)
Given a higher tensor, we construct a sequence of multilinear free resolutions. The construction is entirely explicit and simultaneously generalizes the construction of the Eagon--Northcott complex, the Buchsbaum--Rim complex, the Gelfand--Kapranov--Zelevinsky hyperdeterminantal complexes, and the Eisenbud--Schreyer pure resolutions.
Poset structures in Boij-Söderberg theory,
joint w/ Christine Berkesch, Manoj Kummini, and Steven V Sam.
( PDF or
arXiv: 1010.2663.)
We illustrate that the existence of morphisms between supernatural sheaves (resp. modules with pure resolutions) can be used to recover the poset structure on the cone of cohomology tables (resp. the cone of Betti diagrams). Our results strongly suggest the naturality of the poset structures, and they provide tools for extending Boij-Söderberg theory to other graded rings
and projective varieties.
Secant Varieties of P^2 x P^n embedded by O(1,2),
joint w/ Dustin Cartwright and Luke Oeding.
( PDF or
arXiv: 1009.1199.)
We describe the defining ideal of the rth secant variety of P^2 x P^n embedded by O(1,2), for arbitrary n and r at most 5. We also present the Schur module decomposition of the space of generators of each such ideal. This extends previous work of Strassen and Ottaviani.
Beyond Numerics: The Existence of Pure Filtrations,
joint w/ David Eisenbud and
Frank Olaf-Schreyer.
( PDF or
arXiv: 1001.0585.)
We consider the question of whether the Boij-Söderberg theoeretic decomposition of a Betti diagram into a rational linear combination of pure diagrams ever corresponds to an actual filtration of the minimal free resolution itself. Our main result is an affirmative answer to this question in many suprising cases. We also obtain new results about the semigroup of Betti diagrams and about very singular spaces of matrices.
Laurent polynomials and Eulerian numbers,
joint w/ Gregory G. Smith and
Anthony Várilly-Alvarado.
(To appear in the Journal of Combintorial Theory, Series A. PDF or
arXiv: 0908.2609.) Prompted by a question of Sturmfels, we show a surprising connection
between Laurent polynomials
and Eulerian numbers. The proof involves reinterpreting the problem
in terms of toric geometry.
A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture.
(Mathematical Research Letters, Vol. 17, No. 6, pp. 1079-1089 (2010).
PDF or
arXiv: 0902.0316.)
We prove a special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture via Boij-Söderberg
theory. More specifically, we show that the
conjecture holds for graded modules where the regularity of
M is small relative to the minimal degree of a first syzygy of M.
A syzygetic approach to the smoothability of 0-schemes of regularity two,
joint w/ Mauricio Velasco. (Advances in Mathematics, Volume 224, Issue 3, 20 June 2010, Pages 1143--1166.
PDF or
arXiv: 0812.3342.)
In this paper we introduce a syzygetic invariant which induces necessary conditions for the smoothability of a 0-scheme. In low degree, these conditions are sufficient for smoothability as well.
The Semigroup of Betti Diagrams.
(Algebra and Number Theory, Vol. 3 (2009), No. 3, 341-365.
PDF or
arXiv: 0806.4401.)
We investigate the semigroup of Betti diagrams and answer several fundamental questions about this semigroup, such as a proof that the semigroup is finitely generated.
Hilbert schemes of 8 points,
joint w/ Dustin A. Cartwright, Mauricio Velasco and Bianca Viray.
(Journal of Algebra and Number Theory, Vol. 3 (2009), No. 7, 763--795.
PDF or
arXiv: 0803.0341.)
We classify the irreducible components and the intersection locus of Hilbert schemes of at most 8 points in affine space.
Expository Odds and Ends
Open Questions in Algebraic Geometry. (PDF.) These are collected notes from a session on open questions at the MSRI semester on algebraic geometry in Spring 2009. These notes are an unofficial record of the discussion, and we make no guarantees of accuracy.
Open Questions Related to Boij-Söderberg Theory, (PDF.) These are collected from notes during the 2008 conference on Minimal Free Resolutions at Cornell University.
Frank Moore and Joanna Nilsson helped me edit this document.
NOTE: Progress has been made on some of these questions since 2008.
If you're curious about the state of the art, feel free to email.
The CME Project
(Website.) I was one of many authors for this NSF-funded high school curriculum. The lead developer of the project was Al Cuoco.
Together with
Irena Swanson
and
Amelia Taylor,
I am co-organizing a summer school on commutative algebra at MSRI for 2011.
More information can be found
here.
MathOverflow
I'm a fan of
MathOverflow.net,
which is a site for asking and answering questions about research mathematics.