In my research, I seek to identify and exploit combinatorial
structure in algebraic geometry, often via a group action on a
variety. For instance, polyhedral geometry and certain simplicial
fans characterize the ranks of cohomology of vector bundles on
projective space. Dually, over a standard graded polynomial ring,
combinatorial structure governs the numerics of free resolutions,
homological objects that contain a wealth of geometric data. I am
interested in applications and generalizations of this framework,
which is called Boij--Söderberg theory.

In another setting, combining lattice point geometry with Koszul
homology yields a formula for the dimension of the analytic
solution space of A-hypergeometric systems, certain systems of
PDEs that arise naturally from a torus action. I study properties
of these systems and seek to generalize them to the case of
reductive group actions.

**Virtual resolutions for a product of projective spaces**

(with Daniel Erman and Gregory G. Smith)

Given a module M over the Cox ring of a smooth toric variety, one may consider its minimal free resolution, or more broadly, free complexes that are acyclic modulo irrelevant homology. We call such a complex a virtual resolution of M. On projective space, these complexes have essentially no advantage over minimal free resolutions. The main idea in this paper is that classical homological theorems about projective space have strong analogues for smooth toric varieties when minimal free resolutions are replaced by virtual resolutions; we develop this idea for products of projective spaces and propose a list of exciting new questions for the field.

**Torus equivariant D-modules and hypergeometric systems**

(with Laura Felicia Matusevich and Uli Walther)

We formalize, at the level of D-modules, the notion that A-hypergeometric systems are equivariant versions of the classical hypergeometric equations. For this purpose, we construct a functor on a suitable category of torus equivariant D-modules and show that it preserves key properties, such as holonomicity, regularity, and reducibility of monodromy representation. We also examine its effect on solutions, characteristic varieties, and singular loci. When applied to certain binomial D-modules, our functor produces saturations of the classical hypergeometric differential equations, a fact that sheds new light on the D-module theoretic properties of these classical systems.

**On normalized Horn systems**

(with Laura Felicia Matusevich and Uli Walther)

We explain the relationship between a normalized form of a classical Horn hypergeometric system and a binomial Horn system. As applications, we determine when such a classical system is regular and holonomic.

**On the parametric behavior of A-hypergeometric series**

(with Jens Forsgård and Laura Felicia Matusevich)

*Transactions of the AMS,***370**(2018), no. 6, 4089--4109.

We describe the parametric behavior of the series solutions of an A-hypergeometric system. More precisely, we construct explicit stratifications of the parameter spaces such that, on each stratum, the series solutions of the corresponding system are holomorphic.

**Hypergeometric Functions for Projective Toric Curves**

(with Jens Forsgård and Laura Felicia Matusevich)

*Advances in Mathematics*,**300**(2016), 835--867.

We produce a decomposition of the parameter space of the A-hypergeometric system associated to a projective monomial curve as a union of an arrangement of lines and its complement, in such a way that the analytic behavior of the solutions of the system is explicitly controlled within each term of the union.

**Singularities of binomial D-modules**

(with Laura Felicia Matusevich and Uli Walther)

*Journal of Algebra*,**439**(2015), 360--372.

We provide two new characterizations of holonomicity for binomial D-modules (which generalize A-hypergeometric systems). We first describe the singularities of binomial D-modules, and then show that such modules are holonomic if and only if their corresponding singular loci are proper. For the second characterization states the equivalence of holonomicity and L-holonomicity for these systems.

**Systems of parameters and holonomicity of A-hypergeometric systems**

(with Stephen Griffeth and Ezra Miller)

*Pacific Journal of Mathematics*,**276-2**, (2015), 281--286.

We provide an elementary proof of holonomicity for A-hypergeometric systems, with no requirements on the behavior of their singularities, originally due to Adolphson, after the regular singular case by Gelfand and Gelfand. Our method yields a direct de novo proof that A-hypergeometric systems form holonomic families over their parameter spaces, as shown by Matusevich, Miller, and Walther.

**Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k+1)-equals ideal**

(with Stephen Griffeth and Steven V Sam)

*Communications in Mathematical Physics*,**330**(2014), no. 1, pp. 415--434.

We show that for Jack parameter \alpha = -(k+1)/(r-1), certain Jack polynomials studied by Feigin-Jimbo-Miwa-Mukhin vanish to order r when k+1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. We also conjecture that unitary representations of the type A Cherednik algebra have graded minimal free resolutions of Bernstein-Gelfand-Gelfand type; we prove this for the ideal of the (k+1)-equals arrangement in the case when the number of coordinates n is at most 2k+1.

**Euler--Mellin integrals and multivariate hypergeometric functions**

(with Jens Forsgård and Mikael Passare)

*Michigan Mathematical Journal*,**63**, no. 1, 101--123.

We provide a meromorphic continuation of integrals that generalize the Mellin transform of a rational function 1/f and classical Euler integrals. The components of the complement of the closed coamoeba give a family of these integrals, which we then apply to the theory of A-hypergeometric functions.

**Tensor complexes: Multilinear free resolutions constructed from higher tensors**

(with Daniel Erman, Manoj Kummini, and Steven V Sam)

*Journal of the European Mathematical Society*,**15**(2013), no. 6, 2257--2295.

We explicitly construct a family of minimal free resolutions a higher tensor, providing a unifying view on a wide variety of complexes including: the Eagon--Northcott, Buchsbaum--Rim and similar complexes, the Eisenbud--Schreyer pure resolutions, and the complexes used by Gelfand--Kapranov--Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij--Söderberg theory, including the construction of infinitely many new families of pure resolutions, and the first explicit description of the differentials of the Eisenbud--Schreyer pure resolutions.

**Three flavors of extremal Betti tables**

(with Daniel Erman and Manoj Kummini)

*Commutative Algebra*, Springer, ed. Irena Peeva (2013), 99--122.

We discuss extremal Betti tables of resolutions in three different contexts. We begin over the graded polynomial ring, where extremal Betti tables correspond to pure resolutions. We then contrast this behavior with that of extremal Betti tables over regular local rings and over a bigraded ring.

**Poset structures in Boij--Söderberg theory**

(with Daniel Erman, Manoj Kummini, and Steven V Sam)

*International Mathematics Research Notices*(2012), vol. 2012, 5132--5160.

Boij--Söderberg theory is the study of two cones: the cone of cohomology tables of coherent sheaves over projective space and the cone of standard graded minimal free resolutions over a polynomial ring. Each cone has a simplicial fan structure induced by a partial order on its extremal rays. We provide a new interpretation of these partial orders in terms of the existence of nonzero homomorphisms. These results provide new insights into certain families of supernatural sheaves and Cohen--Macaulay modules with pure resolutions, suggest the naturality of these partial orders, and provide tools for extending Boij--Söderberg theory to other graded rings and projective varieties.

**Shapes of free resolutions over a local ring**

(with Daniel Erman, Manoj Kummini, and Steven V Sam)

*Mathematische Annalen*,**354**(2012), no. 3, 939--954.

We classify the possible shapes of minimal free resolutions over a regular local ring, showing the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal \QQ-Betti sequences.

**The cone of Betti diagrams over a hypersurface ring of low embedding dimension**

(with Jesse Burke, Daniel Erman, and Courtney Gibbons)

*Journal of Pure and Applied Algebra*,**216**(2012), 2256--2268.

We provide the first example of a graded ring besides the standard graded polynomial ring for which the cone of Betti diagrams is entirely understood.

**The rank of a hypergeometric system**

*Compositio Mathematica*,**147**(2011), no. 1, 284--318.

An A-hypergeometric system is the D-module counterpart of a toric ideal: it is a system of linear partial differential equations determined by an integer matrix and a complex parameter vector, and its solutions occur naturally in mathematics and physics. We derive a formula for the dimension of its solution space at any parameter, greatly improving our understanding of the parametric behavior of its solutions.

**Algorithms for Bernstein--Sato polynomials and multiplier ideals**

(with Anton Leykin)

*ISSAC 2010*.

The Bernstein--Sato polynomial (or global b-function) is an important invariant in singularity theory. We develop a new method to compute its local version and develop algorithms for the generalized b-functions of Budur--Mustata--Saito and Shibuta. We then use these to simplify Shibuta's algorithm for computing multiplier ideals.

**A-graded methods for monomial ideals**

(with Laura Felicia Matusevich)

*Journal of Algebra*,**322**(2009), 2886--2904.

We provide topological and enumerative tests for the Cohen--Macaulay property of an arbitrary monomial ideal. The first is a generalization of Reisner's criterion, while the second is given by an explicit intersection multiplicity formula.

Expository works:

**Syzygies, finite length modules, and random curves**(with Frank-Olaf Schreyer), 2013.

**Boij--Söderberg theory and tensor complexes**,*Oberwolfach Report*(extended abstract), Mini-Workshop: Constructive Homological Algebra with Applications to Coherent Sheaves and Control Theory, May 2013.

The seminars I attend most often at Minnesota:

All UMN seminars
I am an organizer of CA+. (This was formerly the Upper Midwest
Commutative Algebra Colloquium.)

I was an organizer of Local
cohomology in commutative algebra and algebraic geometry on the
occasion of Gennady Lyubeznik's 60th birthday.

Fall 2018:

- Math 8201 General Algebra

- Math 8300 Topics

- Math 8212 Commutative and Homological Algebra

- Math 5385 Introduction to Computational Algebraic Geometry

- Math 8211 Commutative and Homological Algebra

- Math 8202 General Algebra

- Math 1271 Calculus I

- Math 8212 Commutative and Homological Algebra

- Math 8211 Commutative and Homological Algebra

- Math 4281 Introduction to Modern Algebra

- Math 4281 Introduction to Modern Algebra

I am the faculty advisor for Women
In Mathematics at the University of Minnesota and the
EDGE Program's Minnesota
Mentoring Cluster.

I helped begin the program Math Ment♀ring at
Duke University.
It is aimed at female undergraduates interested in a
mathematics major or minor.
It is a part of a mentoring pyramid of female mathematicians, which
includes the
Noetherian
Ring, postdocs, and faculty at Duke.

**Mathematical Community Links**

- AMS: American Mathematical Society
- MAA: Mathematical Association of America
- SI(AG)^2: SIAM activity group in Algebraic Geometry
- AMS MRC: AMS Mathematical Research Communities
- www.commalg.org

**Mathematical Resources**

- MathSciNet
- The arXiv
- mathoverflow: A Mathematical Q&A Page

**Computer algebra software that I have used in my research**

**Promoting women in science and mathematics!**

- AWM: Association for Women in Mathematics
- The EDGE Program: Enhancing Diversity in Graduate Education
- MathMentoring at Duke
- WISP: Purdue Women in Science Program