Interests:

My research interests are in algebraic, geometric, and topological combinatorics, including:

• Integer points in rational polytopes and rational cones
• Quasi-polynomials and generating functions arising from enumerative problems
• Integer partitions and compositions
• Topological properties of partially ordered sets
• Simplicial complexes arising via combinatorial constructions
• Cellular resolutions of monomial ideals
• Groebner bases for toric ideals

I am also interested in scholarship regarding teaching and learning mathematics, including:

• Using writing in mathematics courses
• K-12 teacher education
• Pedagogical use of the history of mathematics
• Connections between active learning methods and cognitive/social/educational psychology

Funding:

• National Science Foundation award DMS-0758321, 2008-2012.
• National Security Agency Young Investigators Grant H98230-13-1-0240, 2013-2015.
• National Security Agency Young Investigators Grant H98230-16-1-0045, 2016-2018.

Publications:

(Scholarship of teaching and learning publications follow the research publication list.)

Research:

1. Rationality of Poincare Series for a Family of Lattice Simplices, (joint with Brian Davis)
submitted.
• We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincar\'e series is rational. Our method of proof is to produce for each algebra an explicit minimal free resolution of the ground field, in which the resolution reflects the recursive structure encoded in the denominator of the finely-graded Poincar\'e series. Using this resolution, we show that these algebras are not Koszul, and therefore rationality is non-trivial. Our results demonstrate how interactions between multivariate and univariate rational generating functions can create subtle complications when attempting to use rational Poincar\'e series to inform the construction of minimal resolutions.

2. Laplacian Simplices, (joint with Marie Meyer)
submitted.
• This paper initiates the study of the "Laplacian simplex" $$T_G$$ obtained from a finite graph $$G$$ by taking the convex hull of the columns of the Laplacian matrix for $$G$$. Basic properties of these simplices are established, and then a systematic investigation of $$T_G$$ for trees, cycles, and complete graphs is provided. Motivated by a conjecture of Hibi and Ohsugi, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart $$h^*$$-vectors. We prove that if $$G$$ is a tree, odd cycle, complete graph, or a whiskering of an even cycle, then $$T_G$$ is reflexive. We show that while $$T_{K_n}$$ has the integer decomposition property, $$T_{C_n}$$ for odd cycles does not. The Ehrhart $$h^*$$-vectors of $$T_G$$ for trees, odd cycles, and complete graphs are shown to be unimodal. As a special case it is shown that when $$n$$ is an odd prime, the Ehrhart $$h^*$$-vector of $$T_{C_n}$$ is given by $$(h_0^*,\ldots,h_{n-1}^*)=(1,\ldots,1,n^2-n+1,1,\ldots, 1)$$. We also provide a combinatorial interpretation of the Ehrhart $$h^*$$-vector for $$T_{K_n}$$.

3. Counting Arithmetical Structures on Paths and Cycles, (joint with Hugo Corrales, Scott Corry, Luis David Garcia Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. Valencia)
submitted.
• Let $$G$$ be a finite, simple, connected graph. An arithmetical structure on $$G$$ is a pair of positive integer vectors $$d,r$$ such that $$(diag(d)-A)r=0$$, where $$A$$ is the adjacency matrix of $$G$$. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the cokernels of the matrices $$(diag(d)-A))$$. For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients $${2n-1 \choose n-1}$$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

4. Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices, (joint with Robert Davis and Liam Solus)
submitted.
• The Ehrhart series of a lattice polytope $$P$$ is a rational function encoding the number of lattice points in nonnegative integer scalings of $$P$$. The numerator of this series is the (Ehrhart) $$h^\ast$$-polynomial of $$P$$. An active topic of research is to characterize those polytopes for which the distribution of the coefficients of the $$h^\ast$$-polynomial is unimodal. It was originally conjectured that all reflexive polytopes have unimodal $$h^\ast$$-polynomials. However, counterexamples to this conjecture were found in the form of simplices. A weaker, and currently wide-open, conjecture claims that all Gorenstein polytopes with the integer decomposition property (IDP) have unimodal $$h^\ast$$-polynomials. Based on the counterexamples to the original conjecture, it is worthwhile to investigate the validity of the open conjecture in the special case of reflexive simplices. The collection of reflexive simplices admits a classification in terms of arithmetic sequences. In this paper, we use this arithmetic classification to recast the open conjecture in the language of number theory. We first provide a number theoretic characterization of the $$h^\ast$$-polynomials and IDP for a subfamily of reflexive simplices. We then develop a systematic framework by which to study this problem and validate the conjecture for families of reflexive simplices within this setting. Evenmore, we see there exist simplices within these families that meet only a necessary (but not sufficient) condition for IDP that also have unimodal $$h^\ast$$-polynomials.

5. Euler-Mahonian Statistics and Descent Bases for Semigroup Algebras, (joint with McCabe Olsen)
submitted.
• The Euler-Mahonian identity is a bivariate generalization of the Eulerian polynomial identity using the joint distribution of the descent number and major index over the symmetric group $$S_n$$. We produce new proofs of this identity and generalizations over colored permutation groups $$\mathbb{Z}_r\wr S_n$$ by considering descent bases of quotients of the unit cube semigroup algebra. In doing so, we provide a new algebraic interpretation of the negative descent and negative major index statistics for $$\mathbb{Z}_r\wr S_n$$.

6. Matching and Independence Complexes Related to Small Grids, (joint with Wesley K. Hough)
Electron. J. Comb., 24(4) (2017), #P4.18
• The topology of the matching complex for the $$2\times n$$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $$\mathrm{Ind}(\Delta_n^m)$$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $$\mathrm{Ind}(\Delta_n^m)$$. Further, we determine the Euler characteristic of $$\mathrm{Ind}(\Delta_n^m)$$ and prove that several homology groups of $$\mathrm{Ind}(\Delta_n^m)$$ are non-zero.

7. Generating Functions and Triangulations for Lecture Hall Cones, (joint with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos)
SIAM J. Discrete Math., 30(3), 1470-1479.
• We investigate the arithmetic-geometric structure of the lecture hall cone $L_n \ := \ \left\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq \frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq \frac{\lambda_n}{n}\right\} .$ We show that $$L_n$$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $$h^*$$-polynomial is given by the $$(n-1)$$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $$L_n$$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $$L_n$$, including connections between enumerative and algebraic properties of $$L_n$$ and cones over unit cubes.

8. Unimodality Problems in Ehrhart Theory
in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27
• Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $$h^*$$-vector. Ehrhart $$h^*$$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $$h^*$$-vectors and highlight open questions and problems.

9. Shellability, Ehrhart Theory, and $$r$$-stable Hypersimplices, (joint with Liam Solus)
submitted.
• Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called $$r$$-stable hypersimplices, and show that a standard unimodular triangulation of the hypersimplex restricts to a unimodular triangulation of each $$r$$-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an $$n$$-set, we provide a shelling of this triangulation that sequentially shells each $$r$$-stable sub-hypersimplex. In this case, we also investigate connections between the Ehrhart h*-vector of the second hypersimplex and the $$r$$-stable sub-hypersimplices.

10. Ehrhart series, unimodality, and integrally closed reflexive polytopes, (joint with Robert Davis)
Ann. Comb. 20 (2016), no. 4, 705-717.
• An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h*-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive polytopes and discuss an operation that preserves reflexivity, integral closure, and unimodality of the h*-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the special case of reflexive simplices.

11. Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes, (joint with Sarah Crown Rundell)
Electron. J. Comb., 21(2) (2014), #P2.35
• Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph $$G$$ are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for $$G$$. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.

12. s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones, (joint with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos)
The Ramanujan Journal, February 2015, Volume 36, Issue 1-2, pp 123-147.
• In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $$((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $$((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research.

13. Lattice Point Generating Functions for Symmetric Cones, (joint with Matthias Beck, Thomas Bliem, and Carla Savage)
Journal of Algebraic Combinatorics 38 (2013), 543-566.
• We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.

14. Compositions constrained by graph Laplacian minors, (joint with Robert Davis, Jessica Doering, Ashley Harrison, Jenna Noll, and Clifford Taylor),
INTEGERS 13 (2013), paper A41.
• Motivated by examples of symmetrically constrained compositions, super convex partitions, and super convex compositions, we initiate the study of partitions and compositions constrained by graph Laplacian minors. We provide a complete description of the multivariate generating functions for such compositions in the case of trees. We answer a question due to Corteel, Savage, and Wilf regarding super convex compositions, which we describe as compositions constrained by Laplacian minors for cycles; we extend this solution to the study of compositions constrained by Laplacian minors of leafed cycles. Connections are established and conjectured between compositions constrained by Laplacian minors of leafed cycles of prime length and algebraic/combinatorial properties of reflexive simplices.

15. Euler-Mahonian Statistics via Polyhedral Geometry, (joint with Matthias Beck),
Advances in Mathematics 244 (2013), 925-954.
• A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to pairs of such statistics is an Euler--Mahonian distribution, a bivariate generating function identity encoding these statistics. We use techniques from polyhedral geometry to establish new multivariate generalizations for many of the known Euler--Mahonian distributions. The original bivariate distributions are then straightforward specializations of these multivariate identities. A consequence of these new techniques are bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

16. Mahonian Partition Identities via Polyhedral Geometry, (joint with Matthias Beck and Nguyen Le),
From Fourier Analysis and Number Theory to Radon Transforms and Geometry: In Memory of Leon Ehrenpreis, (H. Farkas, R. Gunning, M. Knopp, and B. A. Taylor, eds.), Developments in Mathematics 28 (2013), 41--54.
• In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $$\Omega$$ operator to systematically compute generating functions for some set of integer partitions. Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.

17. Cellular Resolutions of Ideals Defined by Simplicial Homomorphisms, (joint with Jonathan Browder and Steven Klee),
Israel J. Math. 196 (2013), no. 1, 321-344. DOI: 10.1007/s11856-012-0149-2.
• In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. As a key ingredient of our work, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.

18. Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs, (joint with Matthew Zeckner),
Proc. Amer. Math. Soc. 142 (2014), 413-427.
• In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph $$SG_{n,k}$$ is homotopy equivalent to a k-sphere. Further, for $$n=2$$ they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of $$SG_{n,k}$$ contains as a deformation retract the boundary complex of a simplicial polytope. Our purpose is to give a positive answer to this question in the case $$k=2$$. We also find in this case that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of $$SG_{n,2}$$.

19. Independence Complexes of Stable Kneser Graphs,
Electronic Journal of Combinatorics, 18, no. 1 (2011), P118.
• For integers $$n \geq 1, k \geq 0,$$ the stable Kneser graph $$SG(n,k)$$ (also called the Schrijver graph) has as vertex set the stable n-subsets of $$[2n + k]$$ and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form $$\{i, i + 1\}$$ or $$\{1, 2n + k\}$$. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number k+2. This article contains a study of the independence complexes of $$SG(n,k)$$ for small values of n and k. Our contributions are two-fold: first, we find that the homotopy type of the independence complex of $$SG(2,k)$$ is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to $$SG(n,2)$$.

20. Nowhere-Harmonic Colorings of Graphs, (joint with Matthias Beck),
Proc. Amer. Math. Soc. 140 (2012), 47-63.
• Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.

21. Symmetries of the Stable Kneser Graphs
Adv. in Appl. Math., 45 (2010), no. 1, 12 - 14.
• It is well known that the automorphism group of the Kneser graph $$KG(n,k)$$ is the symmetric group on n letters, where $$KG(n,k)$$ is the graph whose vertices are the k-subsets of an n-set and edges are given by disjoint k-sets. For $$n \geq 2k + 1, k \geq 2$$, we prove that the automorphism group of the stable Kneser graph $$SG(n,k)$$ is the dihedral group of order 2n.

22. The Complex of Non-Crossing Diagonals of a Polygon, (joint with Richard Ehrenborg)
J. Combin. Theory Ser. A, 117 (2010), no. 6, 642 - 649.
• Given a convex n-gon P in the Euclidean plane, it is well known that the simplicial complex T(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We generalize this result for any non-convex polygonal region P with n vertices and h+1 boundary components. We also provide a new proof that T(P) is a sphere when P is convex.

23. Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem, (joint with Mike Develin)
Integer Points in Polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, Contemporary Mathematics 2008, Volume: 452, pp 67-78.
• Stanley's non-negativity theorem is at the heart of many of the results in Ehrhart theory. In this paper, we analyze the root behavior of general polynomials satisfying the conditions of Stanley's theorem and compare this to the known root behavior of Ehrhart polynomials. We provide a possible counterexample to a conjecture of the second author, M. Beck, J. De Loera, J. Pfeifle, and R. Stanley, and contribute some experimental data as well.

24. Norm Bounds For Ehrhart Polynomial Roots,
Discrete and Computational Geometry, 39 (2008), no. 1-3, 191-193.
• M. Beck, J. De Loera, M. Develin, J. Pfeifle and R. Stanley found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.

25. An Ehrhart Series Formula For Reflexive Polytopes,
Electronic Journal of Combinatorics, 13, no. 1 (2006), N 15.
• This paper contains a proof that the numerator of the Ehrhart series for the free sum of a reflexive polytope and another lattice polytope containing the origin is the product of the numerators for the Ehrhart series of the summands.

26. Ehrhart Theory for Lattice Polytopes, my PhD thesis.
• My thesis is a combination of the papers "Norm Bounds...," "An Ehrhart Series Formula...," and "Ehrhart Polynomial Roots..." shown above.

Scholarship of Teaching and Learning:

I am the Editor-in-Chief of the American Mathematical Society blog On Teaching and Learning Mathematics. Some of my contributions to the blog include:

SoTL Publications:
1. MAA Instructional Practices Guide, Steering Committee member and lead writer for chapter on assessment. MAA, 2017.

2. What Does Active Learning Mean For Mathematicians?, joint with Priscilla Bremser, Art M. Duval, Elise Lockwood, and Diana White
Notices of the American Mathematical Society, February 2017, Volume 64, Number 2, 124--129

3. Learning Proofs in the Hallway
MAA Focus, April/May 2016, pgs 32-33

4. Persistent Learning, Critical Teaching: Intelligence Beliefs and Active Learning in Mathematics Courses
Notices of the American Mathematical Society, January 2014, Volume 61, Number 1, 72--74
• This article highlights contemporary research in social psychology and mathematics education with the purpose of identifying connections between beliefs about the nature of intelligence and active learning methods.

5. Personal, Expository, Critical, and Creative: Using Writing in Mathematics Courses
(a previous version of this paper containing more quotes from students is available here)
PRIMUS (Problems, Resources, and Issues in Undergraduate Mathematics Studies), 24 (6), 2014, 447-464.
• This article provides a framework for creating and using writing assignments based on four types of writing: personal, expository, critical, and creative. This framework identifies not only these four types of writing, but also the areas of student growth they can be used to address. Illustrative sample assignments are given throughout for each of these types of writing, and various combinations thereof. Also discussed are the assessment of mathematical writing, the complimentary role played by student reading of mathematical texts, and suggestions for beginning users of writing in mathematics courses.