

Teaching (Spring 2018)
An abstract reasoning course, highlighting the
connections between college mathematics and high school algebra and precalculus. For past teaching at UNL click here. All course materials can be found on Blackboard or Canvas. I am a faculty advisor for the UNL Math Club and the organizer
of the new lecture series Career Perspectives in
Mathematics. In April 2018 I am organizing the Central States Mathematics
Undergraduate Research Conference (CeSMUR
2018) at UNL. Recent and
upcoming conferences
Research and
Publications
My research is in commutative algebra, with an interest
in computational algebra, homological methods and connections to algebraic
geometry. 1.
Computations
involving symbolic powers (with B. Drabkin, E. Grifo, B. Stone) and the Macaulay2
package SymbolicPowers Symbolic powers
are a classical commutative algebra topic that relates to primary
decomposition, consisting, in some circumstances, of the functions that
vanish up to a certain order on a given variety. However, these are
notoriously difficult to compute, and there are seemingly simple questions
related to symbolic powers that remain open even over polynomial rings. In
this paper, we describe a Macaulay2 software package that allows for
computations of symbolic powers of ideals and which can be used to study the
equality and containment problems, among others. 2. Negative curves on symmetric blowups
of the projective plane, resurgences and Waldschmidt
constants (with Th. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, T. Szemberg) In this paper
we study the surface X obtained by blowing up the projective plane in the
singular points of one of two highly symmetric line configurations known as
the Klein configuration and the Wiman configuration
respectively. We study invariant curves on X in detail, with a particular emphasis
on curves of negative selfintersection. We use the representation theory of
the stabilizers of the singular points to discover several invariant curves
of negative selfintersection on X, and use these curves to study Nagatatype
questions for linear series on X. 3. Symbolic powers of codimension two CohenMacaulay ideals (with S. Cooper, G. Fatabbi, E. Guardo, A. Lorenzini, J. Migliore, U.
Nagel, J. Szpond, A. Van Tuyl) Under the additional
hypothesis that X is a local complete intersection, we classify when I(X)^(m) = I(X)^m for all m >= 1. The key tool to prove
this classification is the ability to construct a graded minimal free
resolution of I^m under these hypotheses. Among our
applications are significantly simplified proofs for known results about
symbolic powers of ideals of points in P^1 x P^1. 4.
The Waldschmidt
constant for squarefree monomial ideals (with C. Bocci, S. Cooper, E. Guardo, B.
Harbourne, M. Janssen, U. Nagel, A. Van Tuyl, T. Vu) J. Algebraic
Combinatorics 44 (2016) no. 4, 875904 Given a squarefree monomial ideal I, we show that the Waldschmidt constant of I can be expressed as the optimal
solution to a linear program constructed from the primary decomposition of I.
By applying results from fractional graph theory, we can then express the Waldschmidt constant in terms of the fractional chromatic
number of a hypergraph also constructed from the primary decomposition of I. Moreover, we prove a Chudnovskylike
lower bound on this constant, thus verifying a conjecture of CooperEmbreeHaHoefel for monomial
ideals in the squarefree case.
J. Pure Applied Algebra 219 (2015) no.11, 48574871 We establish a
criterion for the failure of the containment of the symbolic cube in the
square for 3generated ideals I defining reduced sets of points in P^2. Our
criterion arises from studying the minimal free resolutions of the powers of
I, specifically the minimal free resolutions for I^2 and I^3. We apply this criterion
to two point configurations that have recently arisen as counterexamples to a
question of B. Harbourne and C. Huneke:
the Fermat configuration and the Klein configuration. 6.
Determinants of incidence and Hessian
matrices arising from the vector space lattice (with S. Nasseh and J. Watanabe) J. Commutative Algebra We give
explicit formulas for the determinants of the incidence and Hessian matrices
arising from the interaction between the rank 1 and rank n−1 level sets
of the subspace lattice of an ndimensional finite vector space. Our
exploration is motivated by the fact that both of these matrices arise
naturally in the study of the combinatorial and algebraic Lefschetz
properties. 7.
Resurgences for ideals of
special point configurations in P^N coming from hyperplane arrangements (with M. Dumnicki, B. Harbourne, U.
Nagel, T. Szemberg and H. TutajGasińska )
J. Algebra 443 (2015), 383394 It had been expected
for several years that I^(Nr−N+1)⊆ I^r should hold for the ideal I of any finite
set of points in P^N for all r>0, but in the last year various
counterexamples have now been constructed, all involving point sets coming
from hyperplane arrangements. In the present work, we compute their
resurgences and obtain in particular the first examples where the resurgence
and the asymptotic resurgence are not equal. 8.
A tight bound on the projective
dimension of four quadrics (with
C. Huneke, P. Mantero and
J. McCullough) with accompanying Macaulay 2 code. 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 6. 9. A
multiplicity bound for graded rings and a criterion for the CohenMacaulay
property (with
C. Huneke, P. Mantero and
J. McCullough) Proc.
Amer. Math. Soc. 143 (2015) no.6, 23652377 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. 10. Containment
counterexamples for ideals of various configurations of points in P^N
(with B. Harbourne) J. Pure Appl. Algebra 219
(2015), 10621072 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. 11. 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. 12. Multiple
structures with arbitrarily large projective dimension on linear subspaces
(with C. Huneke, P. Mantero
and J. McCullough) J. Algebra 447 (2016), 183205 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. 13. 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. 14. 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). 15. Syzygies and
singularities of tensor product surfaces of bidegree
(2,1) (with H. Schenck
and J. Validashti), Math. Comp. 83 (2014),
13371372 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. 16. 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. 17. 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. 18. 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. 19. 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 and
Videos from Selected Talks
Software
I have
authored or coauthored several packages for the computer algebra program Macaulay2: 

