# Research

Description:

My general interests include algebraic geometry, graph theory, and algebraic combinatorics. Specifically I study the geometric properties and cohomology theories of flag varieties, various domination theories of graphs, stick numbers of knots, and the peak, descent, and pinnacle sets in Coxeter groups. I also advise many student research projects in machine learning, applied mathematics, and data analysis.

Here are some projects I have recently completed or am currently working on:

Pinnacles Sets of Permutations

(Joint with Alexander Diaz-Lopez, Pamela Harris, Lars Nilsen) Fall 2019

(Joint with Alexander Diaz-Lopez, Pamela Harris, Mo Omar, and Bruce Sagan) Fall 2017

Let n be a nonnegative integer and I be a finite set of positive integers. In 1915, MacMahon proved that the number of permutations in the symmetric group S_n with descent set I is a polynomial in n. We call this the descent polynomial. However, basic properties of these polynomials such as a description of their coefficients and roots do not seem to have been studied in the literature. Much more recently, in 2013, Billey, Burdzy, and Sagan showed that the number of elements of S_n with peak set I is a polynomial in n times a certain power of two. Since then, there have been a flurry of papers investigating properties of this peak polynomial. The purpose of the present paper is to study the descent polynomial. We will see that it displays some interesting parallels with its peak relative. Conjectures and questions for future research are scattered throughout.

Computing weight q-multiplicities for the representations of the simple Lie algebras

(Joint with Pamela Harris and Anthony Simpson) Fall 2017

The multiplicity of a weight $\mu$ in an irreducible representation of a simple Lie algebra $\mathfrak{g}$ with highest weight $\lambda$ can be computed via the use of Kostant's weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a q-analog of Kostant's weight multiplicity and present a Sage program to compute $q$-multiplicities for the simple Lie algebras.

The singular locus of semisimple Hessenberg varieties

(Joint with Martha Precup) Fall 2015-Summer 2017

Semisimple Hessenberg varieties are not smooth. In this paper we determine the irreducible components of semisimple Hessenberg varieties corresponding to the standard Hessenberg space. We prove that these irreducible components are smooth and give an explicit description of their intersections, the singular locus. We conclude with examples of semisimple Hessenberg varieties corresponding to other Hessenberg spaces which have singular irreducible components.

Upper broadcast domination of toroidal grids and a classification of diametrical trees

(Joint with Bethany Kubik and Candice Price) Fall 2016

A broadcast on a graph G=(V,E) is a function f:V→{0,1,…,diam(G)} satisfying f(v)≤e(v) for all v∈V, where e(v) denotes the eccentricity of v and diam(G) denotes the diameter of G. We say that a broadcast dominates G if every vertex can hear at least one broadcasting node. The upper domination number is the maximum cost of all possible minimal broadcasts, where the cost of a broadcast is defined as cost(f)=∑v∈Vf(v). In this paper we establish both the upper domination number and the upper broadcast domination number on toroidal grids. In addition, we classify all diametrical trees, that is, trees whose upper domination number is equal to its diameter.

(Joint with Lucas Everham and Vince Marcantonio) Summer and Fall 2016

Since 2012, Billey, Burdzy, and Sagan, and a slew of other authors have studied peak sets of permutations (labelings of path graphs). In this project supported by FGCU's new Seidler Benefaction Undergraduate Research Fellowship, we study peak sets on more general graphs. If G is a graph with n vertices v_1,v_2, ..., v_n, then a permutation of length n corresponds to labeling of the vertices of G. We say that a permutation π has a peak at v_i on G if the label of v_i is greater than all of the labels of v_i's neighboring vertices. (We do not allow peaks at vertices of degree 1 or 0, as these are more like cliffs than peaks!)

A proof of the peak polynomial positivity conjecture

(Joint with Alexander Diaz-Lopez, Pamela Harris, and Mohamed Omar) Spring 2016

**Abstract: **A permutation π = π_1π_2 · · · π_n in the symmetric group S_n has a *peak *at index i if π_{i−1} < π_i > π_{i+1}.

Let P(π) denote the set of indices where has a peak. Given a set S of positive integers, we define P_S(n) ={ π \in S_n:P(π) = S }. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |P_S(n)| = p_S(n) 2^{n-|S|-1} where p_S(x) is a polynomial depending on S. They gave a recursive formula for p_S(x) involving an alternating sum, and they conjectured that the coefficients of p_S(x) expanded in a binomial coefficient basis centered at max(S) are all nonnegative. In this paper we introduce a new recursive formula for |P_S(n)| without alternating sums, and we use this recursion to prove that their conjecture is true.

An Ordered Partition Expansion of the Determinant

(Joint with Katie Johnson and Shaun Sullivan) Fall 2015

**Abstract:** From a transfer formula in multivariate finite operator calculus, comes an expansion for the determinant similar to Ryser's formula for the permanent. Although this one contains many more terms than the usual determinant formula. To prove it, we consider the poset of ordered partitions, properties of the permutahedron, and some good old fashioned combinatorial techniques.

Peak Sets of Classical Coxeter Groups

(Joint with Darleen Perez-Lavin, Pamela Harris, and Alexander Diaz-Lopez) Spring 2015

Accompanying Sage code for this project.

**Abstract: ** A permutation π = π_1π_2 · · · π_n in the symmetric group S_n

has a *peak *at index i if π_{i−1} < π_i > π_{i+1}. Let P(π) = {i | i is a peak of π} be the set of peaks in π. Given a set S of positive integers, we let P(S;n) denote the subset of S_n consisting of all permutations π, where P (π) = S. In 2013, Billey, Burdzy and Sagan proved that |P(S,n)| = p(n)2^{n-|S|-1},

where p(n) is a polynomial of degree max(S) − 1. In 2014, Castro-Velez et, al. considered the Coxeter group of type B as the group of signed permutations on n letters and showed that |PB(S;n)| = p(n)2^{2n−|S|−1}. We embed the Coxeter groups of Lie type B_n and D_n into S_{2n} and use a partitioning of P(S; n) to give a uniform description of the sets of permutations with a given peak set in all classical Coxeter groups.

The pictures above show the Coxeter group of type B_3 and a partition of that group into permutations with a given peak set.

Schubert calculus and the homology of the Peterson variety

Spring 2015

**Abstract:** We use the tight correlation between the geometry of the Peterson variety and the combinatorics of the symmetric group to prove that the homology of the Peterson variety injects into the homology of the flag variety. Our proof counts the points of intersection between certain Schubert varieties in the full flag variety and the Peterson variety and show that these intersections are proper and transverse.

New Upper Bounds on the Distance Domination Numbers of Grids

(Advising Armando Grez and Michael Farina) Fall 2014

**Abstract: ** A subset S of vertices in a graph G is called a *k-*distance domination set if every vertex in G is either in S or it is within *k* edges of a vertex in S. In 2013 Fata, Sundaram, and Smith, developed a grid domination algorithm that established upper bounds for the *k*-distance domination numbers of *m *x *n *grid graphs. Taking a different approach, we use a domination algorithm developed by Blessing, Insko, Johnson, and Mauretour to dominate these grids more efficiently. We then use combinatorial and abstract algebra arguments to prove that these methods give tighter upper bounds for the *k*-distance domination numbers of grids than those established by Fata, Smith, and Sundaram.

On (t,r) Broadcast Domination Numbers of Grids

(Joint with David Blessing, Katie Johnson, and Christie Mauretour) Fall 2013

**Abstract: ** The domination number of a graph G = (V,E) is the minimum cardinality of any subset S of V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem or nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomasse. Many variants of domination number on graphs have been defined and studied, but exact values have not yet been obtained for grids. We will define a family of domination theories parameterized by pairs of positive integers (t,r) where r is between 1 and t which generalize domination and distance domination theories for graphs. We call these domination numbers the (t,r) broadcast domination numbers. We give the exact values of (t,r) broadcast domination numbers for small grids, and we identify upper bounds for the (t,r) broadcast domination numbers for large grids and conjecture that these bounds are tight for sufficiently large grids.

Broadcast domination numbers for grids are hard to predict for small grids. Perhaps the fastest way to find minimal dominating sets for relatively small grids is to crowd-source the problem. Grant Goodman, an undergraduate at FGCU, is developing a web app to allow anyone to dominate grids. If your dominating set is the most efficient to date, then your name will be stored on the leader board!

The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula

(Joint with Pamela Harris and Lauren Kelly Williams) Fall 2013

**Abstract: **Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: *What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra?* We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras of Type B, C, and D, the Type A case was computed by the first author in her Ph.D. thesis. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the Type B case, the cardinality of one such non-zero-weight is enumerated by the Fibonacci numbers. We end with a computational proof of a result of Kostant regarding the exponents of the respective Lie algebra for some low rank examples and provide a section with open problems in this area.

Supercoiled Tangles and Stick Numbers of 2-Bridge knots

(Joint with Rolland Trapp) Summer 2013

The stick number of a knot is the minimum number of sticks needed to make a knot with straight edges in 3-space.

Supercoiling is a phenomenon that occurs naturally in DNA (or your telephone cord). When two strands twist around each other very tightly, they will often supercoil to reduce the tension on the boundary. We show that by allowing integral tangles to supercoil, we can create a knot with essentially 2 sticks for every 3 crossings. This allows us to improve on all previously known bounds by nearly 1/3.

(Joint with Julianna Tymoczko). Fall 2013

A paving by affines of a variety is like a CW-decomposition of a variety X in that it is a decomposition of X into affine open subsets, but the closure relations are not as restrictive as those of a CW-decomposition. Paving by affines are nice because the cells in a paving by affines of an algebraic variety correspond to generators of the homology H_*(X).

We recently proved that regular nilpotent Hessenberg varieties in all Lie types are paved by affines and used this paving to prove geometrically that the cohomology of the flag variety surjects onto the cohomology of Peterson variety in classical Lie types. Discovering this paving by affines is the first step in identifying the equivariant cohomology of regular nilpotent Hessenberg varieties, as it proves that they are equivariantly formal.

Description for non-algebraic geometers:

One can think of what we did as "reverse engineering" the Hessenberg varieties, like the way a chemist decomposes a molecule into constituent atoms. We started with a space, called a Hessenberg variety, that is defined by lots of equations and is very hard to visualize. We decomposed it into simple pieces and kept track of how those pieces glued together, which tells us important properties about the space and gives us a much better idea of what the space really looks like. So discovering a paving by affines for Hessenberg varieties is like saying, "this methane molecule consists of a carbon atom surrounded by four hyrdogen atoms." You can think of the homology of the space as a "cheat sheet" for remembering this sort of information; it tells you how the space decomposes into cells of each dimension.

Patch Ideals and Peterson varieties

(Joint with Alexander Yong.) December 2012

We identify the singular locus of the Peterson varieties and Peterson-Schubert varieties in type A _n using patch ideals. We describe this singular locus via pattern avoidance, and give explicit combinatorial formulas for equivariant K-theory localizations using fundamental facts about projective free resolutions and that regular nilpotent Hessenberg varieties are local complete intersections.

Description for non-algebraic geometers:

An algebraic variety is a space that is cut out by polynomial equations (e.g. the unit circle is cut out by x^2+y^2=1 and a parabola is cut out by y=x^2.) A singularity is point on a variety that "looks different than the rest of the variety." If you look at the picture of the pinched torus above, there is one singular point where the torus is pinched. If you were an ant walking on the surface of the pinched torus the surface would look like a plane (just like the surface of the earth appears flat to us). However, if you were an ant standing on the singular point, then the surface would look like two cones coming together.

The singular locus of an algebraic variety is the collection of all such singular points. Peterson varieties live in higher dimensional space, so we can't just look at them to identify the singular locus.

So instead, we analyze the equations defining them to identify the singular locus. The main result of this paper uses this analysis to give an easy to understand description of the singular locus of the Peterson variety.

Equivariant Cohomology and Local Invariants of Hessenberg Varieties

(My PhD Thesis) August 2012

Nilpotent Hessenberg varieties are a family of subvarieties of the flag variety, which include the Springer varieties, the Peterson variety, and the whole flag variety. In this thesis I give a geometric proof that the cohomology of the flag variety surjects

onto the cohomology of the Peterson variety; I provide a combinatorial criterion for determing the singular loci of a large family of regular nilpotent Hessenberg varieties; and I describe the equivariant cohomology of any regular nilpotent Hessenberg variety whose cohomology is generated by its degree two classes.

Preprints:

Patch ideals and Peterson varieties (with Alexander Yong) preprint (25 pages), January 17, 2011

We study patch ideals of Peterson varieties and some other subvarieties of GL

_{n}/B.

The q-analog of Kostant's partition function and the highest root of the classical Lie algebras

(Joint with Pamela Harris and Mohamed Omar) Fall 2015

The q-analog of Kostant's partition function and the highest root of the classical Lie algebras

(Joint with Pamela Harris and Mohamed Omar) Fall 2015