About
Me
I am an assistant professor
of mathematics at the University of NebraskaLincoln. My research focuses on
commutative algebra with a geometric and computational flavor.
I have been awarded the
2018 Harold & Esther
Edgerton Junior Faculty Award for creative research,
extraordinary teaching abilities, and academic promise.
My Students
PhD students:
 Ben Drabkin – Outstanding First Year
Student award (201516), G.C. and W.H. Young award (20172018)
 Andrew
Conner – SCIENCE
SLAM finalist (2018)
 Erica
Musgrave (coadvised
with Mark Walker)  NPSC fellow (20162021), Outstanding First
Year Student award (201718)
Undergraduate
students:
 Joey
Becker
– Chair’s
prize for best graduating math major (2015)
 Diana
(Xuehua) Zhong – graduate student at North
Carolina State University (since fall 2018)
Undergraduate
Activities
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
 AWM Research Symposium Special Session on Commutative Algebra (coorganized with
Emily Witt), UCLA (April 89 2017)
 CMO Workshop Ordinary and Symbolic Powers of Ideals, Oaxaca, Mexico (May 1419 2017)
 Conference Commutative Algebra
meeting Algebraic Geometry, Bucharest,
Romania (June 2427 2017)
 MittagLeffler Workshop Lefschetz Properties in Algebra, Geometry and
Combinatorics, Djursholm, Sweden
(July 1014 2017)
 Conference and Macaulay2 workshop on Stillman’s
Conjecture and other Progress on Free Resolutions, Berkeley CA (July 1721 2017)
 SIAM Conference on Applied Algebraic Geometry
(AG17), Atlanta (July
31August 4 2017)
 AMS Fall Central Sectional Meeting Special
Session on Commutative Algebra, Denton TX (September 910 2017)
 KUMUNU 2017, Lawrence KS (October 2122 2017)
 Workshop Structures
on Free Resolutions, Lubbock TX (October 2728 2017)
 AMS Spring central Sectional Meeting Special
Session on Lefschetz Properties, Columbus OH (March 1718 2018)
 BIRS workshop New Trends
in Syzygies,
Banff, Canada (June 2429 2018)
 BIRS focused research group on Investigating
Linear Codes via Commutative Algebra (coorganized with S.
Cooper, S. Tohaneanu, A. Van Tuyl), Banff, Canada (July 2229 2018)
 MFO workshop
on Asymptotic
Invariants of Homogeneous Ideals, Oberwolfach, Germany (September 30October
6 2018)
 Sixth conference on Geometric
Methods in Representation Theory, Iowa City IA (November 1719)
 Canadian Math Society Winter Meeting special session
on Symbolic and Regular
Powers of Ideals, Vancouver, Canada (December 89 2018)
 Joint Math Meetings special session Recent
Advances in Homological and Commutative Algebra (coorganized with N.
Epstein and C. Raicu), Baltimore MD (January 1619 2019)
 AWM Research Symposium Special Session on Commutative Algebra, Houston TX (April 67 2019)
Research and
Publications
My research is in commutative algebra, with an interest
in computational algebra, homological methods and connections to algebraic
geometry.
1.
Frieze varieties: A characterization
of the finitetamewild trichotomy for acyclic quivers (with K. Li, L. Li, M. Mills and R. Schiffler)
We introduce a
new class of algebraic varieties which we call frieze varieties. Each frieze
variety is determined by an acyclic quiver. The frieze variety is defined in
an elementary recursive way by constructing a set of points in affine space.
From a more conceptual viewpoint, the coordinates of these points are
specializations of cluster variables in the cluster algebra associated to the
quiver. We give a new characterization of the finitetamewild trichotomy for
acyclic quivers in terms of their frieze varieties by showing that an acyclic
quiver is representation finite, tame, or wild, respectively, if and only if
the dimension of its frieze variety is 0,1, or ≥2,
respectively.
2.
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.
3.
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.
4. 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.
5.
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.
6.
A homological criterion for the failure
of containment of the symbolic cube in the square of some ideals of points in
P^2
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.
7.
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.
8.
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.
9.
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.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. 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).
16. 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.
17. 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.
18. 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.
19. 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.
20. 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
 Resolutions for powers of ideals and applications to
symbolic powers (slides) SIAM
Conference on Applied Algebraic Geometry in Atlanta, Aug 2017
 Combinatorial methods for symbolic powers (video) CMO Workshop on Symbolic
and Ordinary Powers of Ideals in Oaxaca, May 2017
 A handson approach to tensor product surfaces of
bidegree (2,1) (video) CMO Workshop on
Computational Algebra and geometric Modeling in Oaxaca, Aug 2016
 Ordinary and symbolic Rees algebras for Fermat configurations
of points (slides)
AMS
Sectional Meeting in Fargo, Apr 2016
 The Waldschmidt constant for squarefree monomial
ideals (slides)
AMS
Sectional Meeting in Athens, Mar 2016 and AMS Sectional
Meeting in Salt Lake City, Apr 2016
 Polynomial growth for Betti numbers (slides) AMS Sectional Meeting in Athens, Mar 2016
 Using syzygies to test containments between ordinary
and symbolic powers (slides) AMS Joint Mathematics Meeting in San
Antonio, Jan 2015
 Configurations of points and lines with interesting
algebraic properties (slides) AMS Sectional Meeting in Eau Claire, Sept 2014
 Regular vs symbolic powers for ideals of points (slides) AMS Sectional Meeting in Louisville, Oct 2013
 Tight bounds on projective dimension: the case of
quadrics of height two (slides) AMS Sectional Meeting in Louisville, Oct 2013
 The complexity of bounding projective dimension (slides) SIAM
conference on Applied Algebraic
Geometry in Fort Collins, Aug 2013
 Syzygies and singularities of tensor product surfaces
(slides) SIAM
conference on Applied Algebraic
Geometry in Fort Collins, Aug 2013
 A handson approach to tensor product surfaces (slides)
AMS Sectional Meeting in Lawrence, Apr 2012
 Bounding projective dimension and regularity (slides)
AMS Sectional Meeting in Lincoln, Oct 2011
 Fat points and the weak Lefschetz property (slides) International
Congress of Romanian Mathematicians in Brasov, Jul 2011
 From syzygies to the weak Lefschetz property and back (slides)
Resolutions Day at Cornell, May 2011
 Inverse systems, fat points and the weak Lefschetz
property (slides) AMS
Joint Mathematics Meeting in New Orleans, Jan 2011
 Syzygy theorem via comparison of order ideals (slides) AMS
Joint Mathematics Meeting in San Francisco, Jan 2010
 Weak Lefschetz Property and powers of linear forms (slides) AMS
Sectional Meeting in Boca Raton, Nov 2009
 Weak Lefschetz property for ideals generated by
powers of linear forms (poster), PASI conference in Olinda, Aug 2009
Software
I have
authored or coauthored several packages for the computer algebra program Macaulay2:
