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.
I coorganize the Commutative Algebra Seminar at UNL.
My Students
PhD students:
 Ben Drabkin
– Outstanding
First Year Student award (20152016), G.C. and W.H. Young award
(20172018)
 Andrew Conner – twice SCIENCE SLAM finalist
(2018 and 2019)
 Erica
Musgrave (coadvised
with Mark Walker)  NPSC fellow (20162021), Outstanding First
Year Student award (20172018)
 Michael
DeBellevue (coadvised with Mark Walker) – Outstanding
Qualifying Exams award (20182019)
Undergraduate
students:
 Joey
Becker
– Chair’s
prize for best graduating math major (2015)
 Diana
(Xuehua) Zhong – graduate student at North Carolina State
University (since 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 organized the Central States Mathematics
Undergraduate Research Conference (CeSMUR
2018) at UNL. CeSMUR 2019 will take place at Kansas
State University.
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)
 Graduate Workshop in
Commutative Algebra for Women & Mathematicians of Other Minority
Genders, Minneapolis, MN (April 1214 2019)
 Conference on Commutative
Algebra and its Interaction with Algebraic Geometry, South Bend, IN (June 1621 2019)
 Summer
Research for Women in Mathematics program at MSRI, Berkeley, CA (June
23July 6 2019)
 SIAM Conference on Applied Algebraic Geometry
(AG19), Bern, Switzerland
(July 913 2019)
 CIRM workshop on Lefschetz Properties in Algebra, Geometry and
Combinatorics II, Luminy, France
(October 1418 2019)
 BIRS workshop for Women in Commutative Algebra, Banff, Canada (October
2025 2019)
Research and
Publications
My research is in commutative algebra, with an interest
in computational algebra, homological methods and connections to algebraic
geometry.
1. Implicitization
of tensor product surfaces via virtual projective resolutions (with E. Duarte)
We derive the implicit equations for
certain parametric surfaces in threedimensional projective space termed
tensor product surfaces. Our method computes the implicit equation for such a
surface based on the knowledge of the syzygies of the base point locus of the
parametrization by means of constructing an explicit virtual projective
resolution.
2. Betti
numbers of symmetric shifted ideals (with J. Biermann, H. de Alba, F. Galetto,
S. Murai, U. Nagel, A. O’Keefe, T. Römer)
We introduce a new class of monomial ideals
which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by
the natural action of the symmetric group and, within the class of monomial
ideals fixed by this action, they can be considered as an analogue of stable
monomial ideals within the class of monomial ideals. We show that a symmetric
shifted ideal has linear quotients and compute its (equivariant)
graded Betti numbers. As an application of this
result, we obtain several consequences for graded Betti
numbers of symbolic powers of defining ideals of star configurations.
3. Connected sums of graded artinian Gorenstein algebras
and Lefschetz properties (with A. Iarrobino and C.
McDaniel)
A connected sum construction for local
rings was introduced in a paper by H. Ananthnarayan,
L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic
analogue of the topological construction of the same name. We give two
alternative description of this algebraic connected sum: the first uses
algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual
generators. We also investigate the extent to which the connected sum
construction preserves the weak or strong Lefschetz
property.
4. General minimum distance
functions and algebraic invariants of Geramita
ideals (with S. Cooper, Ș.
Tohăneanu, M. Vaz
Pinto and R. Villarreal),
Advances
in Applied Mathematics
Motivated by notions
from coding theory, we study the generalized minimum distance (GMD) function
of a graded ideal I in a polynomial
ring over an arbitrary field using commutative algebraic methods. It is shown
that the GMD function is nondecreasing as a function of its first argument
and nonincreasing as a function of the second argument. For vanishing ideals
over finite fields, we show that the GMD function is in fact strictly
decreasing as a function of the second argument until it stabilizes. We also
study algebraic invariants of Geramita ideals.
Those ideals are graded, unmixed, 1dimensional and their associated primes
are generated by linear forms. We also examine GMD functions of complete
intersections and show some special cases of two conjectures of TohăneanuVan Tuyl and EisenbudGreenHarris.
5. 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.
6. Computations involving
symbolic powers (with B. Drabkin, E. Grifo, B. Stone) and the Macaulay2
package SymbolicPowers
J. Software for
Algebra and Geometry
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.
7.
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) International
Math. Research Notices
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.
8.
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.
9.
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.
10. 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.
11. Determinants
of incidence and Hessian matrices arising from the vector space lattice
(with S. Nasseh and J. Watanabe)
J. Commut. Algebra 11 (2019) no. 1, 131154
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.
12. 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.
13. A tight bound
on the projective dimension of four quadrics (with C. Huneke, P. Mantero and J. McCullough) with accompanying Macaulay 2 code.
J. Pure Appl. Algebra 222 (2018) no. 9, 25242551
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.
14. 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.
15. 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.
16. 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.
17. 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.
18. 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.
19. 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).
20. 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.
21. 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.
22. 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.
23. 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.
24. 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
 Implicitization for tensor product surfaces via virtual projective
resolutions (slides)
SIAM Conference on Applied
Algebraic Geometry in
Bern, Jul 2019
 Generalized minimum distance functions and
applications to coding theory (slides) Canadian
Math Society winter meeting in Vancouver, Dec 2018
 Connected
sums of graded artinian Gorenstein
algebras and the Lefschetz properties (video, slides)
BIRS workshop New Trends in Syzygies in Banff, June 2018
 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:
