

Teaching
Spring 2014 Math 314  Applied Linear Algebra (Matrix Theory) sections 001 and 002
Fundamental
concepts of linear algebra from the point of view of matrix manipulation,
with emphasis on concepts that are most important in applications. Includes
solving systems of linear equations, vector spaces, determinants, eigenvalues, orthogonality and
quadratic forms. For
past teaching at UNL click here. Research
and Publications
My research is in commutative algebra, with an
interest in homological methods and connections to algebraic geometry. 1. Ideals generated by four quadric
polynomials (with
C. Huneke, P. Mantero and
J. McCullough) Motivated by a question posed by Mike Stillman, we show that the projective dimension of an
ideal generated by four quadric forms in a polynomial ring has projective
dimension at most 9. 2. A multiplicity bound for graded rings
and a criterion for the CohenMacaulay property (with C. Huneke, P. Mantero and J. McCullough) We prove an upper bound for the multiplicity of R/I,
where I is a homogeneous ideal of the form I=J+(F)
and J is a CohenMacaulay ideal. The bound is given in terms of invariants of
R/J and the degree of F. We show that ideals achieving this bound have high
depth and deduce a numerical criterion for the CohenMacaulay property.
Applications to quasiGorenstein rings and almost
complete intersection ideals are given. 3. Containment
counterexamples for ideals of various configurations of points in P^N
(with B. Harbourne) to appear in J. Pure Appl.
Algebra We provide counterexamples
to a conjecture of Harbourne and Huneke regarding containments between regular powers and
symbolic powers of ideals of points in projective space P^N. We show that the
conjecture fails in every prime characteristic p>2 when N=2 and we provide
additional counterexamples for higher dimensional projective spaces. 4.
The projective
dimension of codimension two algebra presented by
quadrics (with C. Huneke, P. Mantero and J. McCullough) J. Algebra 393 (2013), 170186 We prove a sharp upper bound for the
projective dimension of ideals of height two generated by quadrics in a
polynomial ring with arbitrary large number of variables. 5. Multiple
structures with arbitrarily large projective dimension on linear subspaces
(with C. Huneke, P. Mantero
and J. McCullough) We show that
no finite characterization of multiple structures on linear spaces is
possible if one only assumes Serre’s S_1 property
holds by by constructing structures with arbitrarily large
projective dimension. Our methods build upon a family of ideals with large
projective dimension using linkage. The result is
in stark contrast to Manolache's characterization
of CohenMacaulay multiple structures in codimension
2 and multiplicity at most 4 and also to Engheta's
characterization of unmixed ideals of height 2 and multiplicity 2. 6. Computations
in intersection rings of flag bundles (with D. Grayson and M. Stillman) This paper arose out of
an observation that was made while I was working on the “Symmetric
polynomials” package for Macaulay2. Intersection rings of flag varieties and
of isotropic flag varieties are generated by Chern
classes of the tautological bundles modulo the relations coming from multiplicativity of total Chern
classes. In this paper we describe the Gröbner
bases of the ideals of relations and give applications to computation of
intersections, as implemented in Macaulay2. 7. Bounding
projective dimension (with J. McCullough) a chapter in the book Commutative Algebra, SpringerVerlag London, 2013 The
celebrated Hilbert Syzygy Theorem states that the
projective dimension of any ideal in a polynomial ring on n variables is at
most n1. This paper surveys recent progress on Stillman’s
question, asking whether the degrees of a set of homogeneous polynomials
suffice in order to bound the projective dimension of the ideal they generate,
without prior knowledge of the ambient polynomial ring (hence without using
the number of variables). 8. Syzygies and singularities of tensor product surfaces of bidegree (2,1) (with H. Schenck and J. Validashti) to
appear in Math. Comp. We study the
associated ideal of a bigraded parametrization
of a surface in P^3 from the standpoint of commutative algebra, proving that
there are exactly six numerical types of possible bigraded
minimal free resolution. These resolutions play a key role in determining the
implicit equation of the image, via work of BuseJouanolou,
BuseChardin, Botbol and BotbolDickensteinDohm on the approximation complex. In
particular this allows us to completely describe the implicit equation and
singular locus of the image. 9. Syzygy theorems via comparison of order ideals on a hypersurface (with P. A. Griffith) J. Pure Appl.
Algebra 216 (2012), no. 2, 468479 We introduce a weak order ideal property that suffices
for establishing the EvansGriffith Syzygy Theorem.
We study this weak order ideal property in settings that allow for comparison
between homological algebra over a local ring R versus a hypersurface
ring R/(x^n). Consequently we solve some relevant
cases of the EvansGriffith syzygy conjecture over
local rings of unramified mixed characteristic p,
with the case of syzygies of prime ideals of
CohenMacaulay local rings of unramified mixed
characteristic being noted. 10. Inverse
systems, fat points and the weak Lefschetz property
(with B. Harbourne and H. Schenck)
J. Lond. Math. Soc. (2) 84 (2011), no. 3,
712730 We use the inverse system dictionary to connect ideals
generated by powers of linear forms to ideals of fat points and show that
failure of WLP for powers of linear forms in at least four variables is
connected to the geometry of the associated fat point scheme. This is in
stark contrast with the situation described in the paper “The Weak Lefschetz Property and powers of linear forms in K[x, y,
z]” below. Closely related results can be found in the paper See the paper "On the weak Lefschetz property for powers of linear forms" by MiglioreMiròRoigNagel. 11. Ideals
with Large(r) Projective Dimension and Regularity (with J. Beder, J. McCullough, L. NunezBetancourt, B. Snapp, B. Stone) J. Symbolic Comput.
46 (2011), no. 10, 11051113 This paper is
an outcome of the Mathematical Research Communities program. I am grateful to
AMS and the organizers for this wonderful opportunity. We define a family of
homogeneous ideals with large projective dimension and regularity relative to
the number of generators and their common degree. This family subsumes and
improves upon constructions given by Caviglia and
McCullough In particular, we describe a family of homogeneous ideals with
three generators of degree d in arbitrary characteristic whose projective dimension
grows asymptotically as. Here is
the Macaulay 2 code mentioned in the paper. 12. The
Weak Lefschetz Property and powers of linear forms in
K[x, y, z] (with H. Schenck) Proc. Amer. Math. Soc. 138 (2010), no. 7,
23352339 We show that any artinian
quotient of K[x, y, z] by an ideal I generated by powers of linear forms has
the Weak Lefschetz Property. Slides from
Selected Talks
Computer
Algebra Projects
I have
authored or coauthored several packages for the computer algebra program Macaulay2: 

