
1. "Stillman's Question for Exterior Algebras and Herzog's Conjecture on Betti Numbers of Syzygy Modules." submitted. [arXiv]
Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional Kvector spaces. In this short paper we exhibit principal quadric ideals in a family whose CastelnuovoMumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that these examples are dual to modules over polynomial rings that yield counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules.

2. "Asymptotically Good Homological Error Correcting Codes."
submitted. (with Heather Newman) [pdf]
Let ∆ be an abstract simplicial complex. We study classical homological error correcting codes associated to ∆, which generalize the cycle codes of simple graphs. It is wellknown that cycle codes of graphs do not yield asymptotically good families of codes. We show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least 2. We also prove general bounds and formulas for (co)cycle and (co)boundary codes for arbitrary simplicial complexes over arbitrary fields.

3. "The Projective Dimension of 3 Cubics is at Most 5." submitted. (with Paolo Mantero) [arXiv]
In this paper we complete work begun by Bahman Engheta in giving an
optimal answer to Stillman's Question in the case of ideals generated
by 3 cubic forms. Engheta had shown previously that any ideal in any
number of variables generated by 3 cubics had projective dimension at
most 36. We give a tight upper bound of 5. The proof is a technical
casebycase analysis and relies heavily on the classification of
(x,y)primary ideals of small multiplicity (see below).

4. "On the Maximal Graded Shifts of Ideals and Modules." submitted. [arXiv]
The inspiration for this paper was the paper "The regularity of
Tor and graded Betti numbers" by EisenbudHunekeUlrich in which the
authors prove several bounds on the maximal graded shifts of modules
of dimension at most one with some assumption about their
annihilator. For example, they show that a the degrees of a
homogeneous complete intersection contained in the annihilator govern
the graded shifts at the end of the resolution. In this paper we
extend these to the case when there is no assumption on the dimension
of the module at the expense of slightly weaker bounds.

5. "Regularity of Prime Ideals."(with Giulio Caviglia, Irena Peeva, Matteo Varbaro) submitted. [pdf]
In this paper we answer a number of natural questions arising from
the counterexamples to the EisenbudGoto Conjecture (see below),
constructed by Peeva and myself; namely, we show (1) There are
counterexamples to the EisenbudGoto conjecture coming from Rees
algebras, (2) there are counterexamples that do not rely on the
MayrMeyer ideals, (3) there is a bound on the regularity (and
projective dimension and graded Betti numbers, etc.) of
nondegenerate prime ideals purely in terms of their multiplicity,
and (4) there exist nondegenerate prime ideals with bounded
generating degree and arbitrarily large regularity.

6. "Counterexamples to the EisenbudGoto regularity conjecture." (with Irena Peeva)
Journal of the American Mathematical Society, Volume 31, Number 2 (2018), 473– 496. [pdf] [link]
There is a general doubly exponential upper bound on the CastelnuovoMumford regularity of homogeneous ideals
in a standard graded polynomial ring. This bound is nearly sharp since the MayrMeyer construction leads to examples of families of ideals
attaining doubly exponential regularity.
It was expected that much better bounds hold for the defining ideals of geometrically nice projective varieties over an algebraically closed field.
The longstanding Regularity Conjecture, by EisenbudGoto (1984), predicts the following elegant linear bound in terms of the degree:
reg(P) ≤ deg(S/P)  codim (P) + 1
for every homogeneous nondegenerate prime ideal P in a standard graded polynomial ring S over an algebraically closed field k.
The Regularity Conjecture holds if S/P is CohenMacaulay by a result of EisenbudGoto (1984).
It is proved for curves by GrusonLazarsfeldPeskine (1983),
completing classical work of Castelnuovo.
It is also proved for smooth surfaces
by Lazarsfeld (1987) and Pinkham (1986),
and for most smooth 3folds by Ran (1990).
In the smooth case, Kwak (1998) gave bounds for regularity in dimensions 3 and 4 that are only slightly worse than the optimal ones in the conjecture.
Many other special cases and related results have been proved as well.
Irena Peeva and I constructed counterexamples to the Regularity Conjecture. Our main theorem is much stronger and shows that the regularity
of nondegenerate homogeneous prime ideals is not bounded by any polynomial function of the degree.
We provide a family of prime ideals
P_{r}, depending on a parameter r, whose degree is singly exponential in r and
whose regularity is doubly exponential in r.
For this purpose, we introduce an approach which, starting from
a homogeneous ideal I, produces a prime ideal P whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The construction involves two new concepts: (1) Reeslike algebras which, unlike the standard Rees algebras, have wellbehaved defining equations and free resolutions, and (2) a stepbystep homogenization technique which, unlike classical homogenization, preserves graded Betti numbers.
A video of David Eisenbud lecturing on our paper at KAIST is available
here: [YouTube link]

7. "A finite classification of (x,y)primary ideals of low multiplicity." (with Paolo
Mantero) Collectanea Mathematica, 69 Issue 1 (2018), 107–130.
[pdf] [link]
Let I be a homogeneous ideal which is primary to a linear prime
(x,y) in a standard graded polynomial ring S. Engheta previously
showed that there are essentially two forms that I can take when the
Hilbert multiplicity of S/I is 2. In previous work with Huneke and
Seceleanu, we showed that for multiplicity 3 and higher there are
infinitely many such ideals and their projective dimension can be
chosen to be arbitrarily high. However, in this paper we give a
finite classification in multiplicities 3 and 4 of the linear,
quadric and cubic generators of all such ideals, showing that there
are essentially 8 cases in multiplicity 3 and 22 cases in
multiplicity 4. We apply our classification theorems to improve
bounds on the projective dimension of 3 cubics  a special case of
Stillman's Question.
[Supplemental M2 File] [Output from Supplemental M2
File]

8. "A tight bound on the projective dimension of four quadrics." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu)
to appear in Journal of Pure and Applied Algebra. [arXiv]
In this paper, we prove an optimal upper bound on the projective dimension
of any homogeneous ideal generated by 4 quadrics in a polynomial
ring over a field. In this specific case, this result vastly improves
the more general upper bound for ideals generated by any number of
quadrics given by AnanyanHochster. [Supplemental M2 File] [Output from Supplemental M2
File]

9. "Three themes on syzygies." (with Gunnar Fløystad and Irena Peeva) Bulletin
of the American Mathematical Society 53 no. 3, July 2016, 415435 [pdf] [link]
This survey paper covers three recent developments in commutative
algebra: resolutions over complete intersections, Stillman's
Question and BoijSoderberg theory. It is intended for a general audience.

10. "Multiple structures with arbitrarily large projective dimension on
linear subspaces." (with Craig Huneke, Paolo Mantero and Alexandra
Seceleanu) Journal of Algebra 447 (2016), 183–205. [arXiv]
[link]
In Engheta's work showing that the projective dimension of any ideal
generated by 3 cubic forms is bounded, he gave a homological
structure theorem for unmixed height2, multiplicity2 ideals. The notable case was
the primary one, in which he showed that the only two possibilities
were (x,y^2) and (x^2,xy,y^2,ax+by), where x,y are independent
linear forms and ht(x,y,a,b) = 4 in the latter case. In this paper
we show that this kind of result cannot by extended by finding
examples of primary ideals with arbitrarily large projective
dimension in every larger height and multiplicity case.

11. "A multiplicity bound for graded rings and a criterion for the
CohenMacaulay property." (with Craig Huneke, Paolo Mantero and Alexandra
Seceleanu) Proceedings of the American Mathematical Society 143
no. 6 (2015), 23652377. [arXiv] [link]
This work extends previous work of Engheta who gave an upper bound
on the HilbertSamuel Multiplicity of a homogeneous almost complete
intersection. Our result gives an upper bound on the multiplicity
of any ideal of the form I + (f), where I is a CohenMacaulay ideal
and f is not in I. Moreover, we show that ideals that meet this
upper bound must be CohenMacaulay themselves. Dual results for
almost Gorenstein ideals are also given as well as several examples
of ideals that meet this upper bound.

12. "Infinite
graded free resolutions." (with Irena Peeva) Commutative Algebra and Noncommutative Algebraic Geometry Volume 1: Expository Articles (Eisenbud, Iyengar, Singh, Stafford, Van den Bergh eds.), Math. Sci. Res. Inst. Publ. No. 67, Cambridge University Press, New York (2015), 215–258. [pdf]
(The version linked here is updated from the print version and
includes references to recent developments in the theory of Golod rings.)
This survey paper is based on Irena Peeva's series of lectures at
the Joint Introductry workshop in Commutative and Cluster Algebras
in Fall 2013 at MSRI. We briefly summarize the construction of
graded free resolutions. The paper covers topics from Koszul rings,
Golod rings, Complete intersections and matrix factorizations, slope, rate and regularity.

13. "Hypergraphs and regularity of squarefree monomial
ideals." (with KueiNuan
Lin)
International Journal of Algebra and Computation 23 no. 7 (2013),
1573–1590. [arXiv] [link]
We define a combinatorial structure which we call a labeled
hypergraph associated to any squarefree monomial ideal. We then
prove a number of special case exact formulas and general upper
bounds on the regularity of I in terms of the combinatorial data in
its labeled hypergraph. Our hypergraph definition is based on one
of KimuraTeraiYoshida who studied the arithmetical rank of certain
monomial ideals.

14. "Projective dimension of codimension two algebras presented by
quadrics." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Journal of Algebra 393 (2013), 170186. [arXiv] [link]
Stillman's Question asks for a bound on the projective dimension of
a homogeneous ideal in a polynomial ring purely in terms of the degrees of the
minimal generators of the ideal. We give an optimal affirmative
answer to this question in the case of height 2 ideals generated by
quadrics.

15. "Bounding projective dimension." (with Alexandra
Seceleanu ) Commutative
Algebra. Ex pository Papers Dedicated to David Eisenbud on the
Occasion of His 65th Birthday (I. Peeva, Ed.), SpringerVerlag,
London (2013), 551576. [pdf]
This is a survey paper written in 2013 on the status of Stillman's
Question. Stillman asked whether the projective dimension of a homogeneous
ideal can be bounded in terms of the degrees of the minimal
generators only. Our paper summarizes Caviglia's proof that
Stillman's question is equivalent to the analogous question for
regularity. We also discuss AnanyanHochster's positive answer for
Stillman's Question in the case of ideals generated by quadrics
along with the construction of ideals of large projective dimension
given below in Beder et. al. Much research has occured since this
was written. Many of the questions we raised have been answered in
the above papers.

16. "A polynomial bound on the regularity of an ideal in
terms of half of the syzygies." Math. Res. Ltrs. 19 (2012), no 3,
555565.[arXiv] [link]
A famous example of MayrMeyer and a modified version of
BayerStillman show that the regularity of an ideal may grow doubly
exponentially with respect to the degrees of the minimal
generators. In his thesis, Engheta asked whether a polynomial upper
bound on regularity would be possible if not just the degrees of the
generators but the degrees of some of the first several syzygies
were used as well. Here I showed that the answer to Engheta's
question is yes if the degrees of the first half of the syzygies are
used. I also prove a related subadditivity upper bound. Both
bounds are proved via the BoijSoederberg decomposition of the Betti
table of the free resolution of R/I.

17. "Ideals
with larger projective dimension and regularity." (with Jesse Beder, Luis NunezBetancourt, Alexandra Seceleanu, Bart Snapp and Branden Stone) J. Symbolic
Comput. 46 (2011), no. 10, 11051113. [arXiv]
[link]
Generalizing the construction below of ideals with large projective
dimension, we construct a family of ideals associated to tuples of
integers that have large projective dimension and regularity
relative to the degrees of the generators. We notably give an
example of a family of 3generated ideals whose projective dimension
grows exponentially with respect to the degrees of the generators.
The generators can be computed easily with the help of the Macaulay2
package BigIdeal.m2. (See below.)

18. "A family of ideals with few generators in low degree and
large projective dimension." Proceedings of the AMS. Volume 139,
No. 6. pp. 20172023. [arXiv] [link]
This is my first paper regarding Stillman's Question in which I
produced a family ideals with large projective dimension relative to
the degrees of the generators. My example showed that a question of
Z. Yi relating to the vanishing of local cohomology modules over a
polynomial ring was false. It also showed bound that any positive answer
to Stillman's Question must necessarily be very large.

19. "A note on the strong direct summand conjecture." Proceedings of
the AMS. Volume 127, No. 9. 2009. pp. 28572864. [pdf]
[link]
Hochster's Direct Summand Conjecture (DSC) and the related Homological
Conjectures have many astonishing consequences in commutative
algebra and algebraic geometry. While most are known to be true,
most are open questions in the mixed characteristic case.
(i.e. where (R,m) is a local ring, char(R) = 0 and char(R/m) = p)
N. Ranganathan gave a new interpretation of one of the homological
conjectures, the Vanishing of Maps of Tor (VMTC), in language similar to
that of the Direct Summand Conjecture. Her Strong Direct Summand
Conjecture (SDSC) is equivalent to the VMTC and implies the DSC. In this
paper, which contains some of the results from my thesis, I prove
several special cases of the SDSC.

20. "SRPT optimally uses faster machines to minimize flow time." (
with Eric Torng ) ACM
Transactions on Algorithms. Volume 5, No. 1. November 2008.
pp. 125. [pdf]
This is research I worked on as an undergraduate with Prof. Eric
Torng at MSU. The problem comes from scheduling theory in computer
science in which one imagines an idealized setting where m parallel
processors must collectively schedule and process jobs of known
length and to minimize flow time (or total waiting time). The
problem is known to be NPhard, but we show that given machines
running the SRPT (Shortest Remaining Processing Time) algorithm with
speed 21/m faster processors than an optimal schedule to which we
are compared, the flow time is 21/m times smaller. This result is
sharp in the sense that for any speedup less than 21/m, there are
schedules for which optimal speed1 machines will outperform the
faster SRPT machines.

21. "A Generalized Floor Bound on the Minimum Distance of Geometric Goppa
Codes." (with Benjamin
Lundell) Journal of Pure and Applied Algebra. Volume 207, Issue 1.
September 2006. pp. 155164. [link]
[pdf]
This work grew out of a summer reading course Ben Lundell and I took
with Prof. Iwan Duursma. We showed how to generalize a bound on the
minimum distance of certain algebraic geometry codes called Goppa
codes in a way that improved the known bounds for certain choices of
divisors on curves over finite fields. We include specific
comparisons of our bound to that of Matthews and KirfelPelikaan for
Suzuki and Hermitian curves. Improvements to our bound have since
been found by GunneriStichtenothTaskin, DuursmaKirovPark, and others.