# Research 2014-2015

My research goals for this year were to continue facilitating interesting student research projects and collaborating with faculty both at FGCU and other institutions. I was also hoping to submit at least one paper to a peer-reviewed journal. I am pleased to say I surpassed these expectations. I submitted a total of two papers to peer-reviewed journals this year, one of which was written with FGCU students. In addition, many of the students who have worked with me this year have presented their work at regional and national conferences. Two of them participated in Research Experiences for Undergraduates last summer, and all three who applied to PhD programs were accepted and funded.

__Summary of student research outcomes:__

A number of the student research collaborations I have worked on over the past two and half years have produced some great results. The students have gained a lot of knowledge and a greater level of mathematical maturity. Here are some of the highlights: Five of my student collaborators presented at our third annual Advancing Student Participation in Research Experiences (ASPiRE) conference, three of them presented at the Joint Mathematics Meetings in San Antonio, one presented at the Emory STEM Symposium Mathematics Conference, one presented at the University of Kentucky Combinatorics Conference, and five of them will present at the Underrepresented Students in Topology and Algebra Research Symposium (USTARS) conference held here at FGCU this April. In addition, Christie Mauretour was accepted into the MAXIMA Research Experience for Undergraduates (REU) 2014 at the University of Minnesota - Twin Cities, and Armando Grez was accepted to the REU at UT-Tyler 2014. Christie Mauretour was inducted into the FGCU Hall of Fame (only 10 students selected from the university's senior class each year).

Most importantly, Darleen Perez-Lavin was accepted into the PhD program at the University of Kentucky with a teaching assistantship; Armando Grez has accepted a prestigious Alliance for Graduate Education and Professoriate (AGEP) fellowship at Iowa State University; and Christie Mauretour was accepted into the graduate program at the University of Florida with a TA ship.

__List of papers published or accepted for publication this year:__

1. Dr. Katie Johnson, David Blessing, Christie Mauretour, and I have published the article on *On (t,r) Broadcast Domination of Grids** *in

the *Journal of Discrete and Applied Mathematics*. To give you an idea of the quality of this publication, this journal is rated as an **A-level **mathematics journal by the Australian Mathematical Society.

2. Dr. Rolland Trapp, a Full Professor at California State University - San Bernardino, and I submitted a paper called *Supercoiled Tangles and Stick Numbers of 2-Bridge Links** *to the *Journal of Knot Theory and its Ramifications** *last spring. (The journal is ranked as a **B-level **mathematics journal by the Australian Mathematics Society.) We recently received notice that the paper was accepted without any suggested revisions and it will appear in print on May 14, 2015.

Here are the referee's flattering comments:

*This is a well written paper concerning an interesting method to construct polygonal models of a very important class of knots and links, the 2-bridge links. Employing a method apparently inspired by DNA conformations, the authors construct highly efficient polygonal tangles that are used as building blocks for the 2-bridge designs and, thereby, give new estimates on the number of edges required in minimal polygonal realizations of these knots and links.*

*I recommend publication of this manuscript in its present form. It is both important and well written. I did not see any places where I would recommend revision.*

3. This spring I submitted a paper *S**chubert calculus and the homology of the Peterson variety** *that resulted from a project that I started in graduate school. In this paper, I use a mix of classical intersection theory and the combinatorics of the symmetric group to prove that the homology of the Peterson variety injects into the homology of the full flag variety.

The paper was recently accepted by the editor of the Electronic Journal of Combinatorics, and will appear in May 2015.

__List of papers recommended for publication this year or under review:__

1. Dr. Julianna Tymoczko of Smith College and I submitted an article *Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety** *to a peer-reviewed algebraic geometry journal last this spring. The article is still under review but was recommended for publication by the referee.

__List of papers I submitted this year:__

1. Dr. Pamela Harris of the US Military Academy at Wespoint, Dr. Lauren Kelly Williams of Mercyhurst University, and I have finished an article on *The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula** *last year. The article is currently under review at a peer-reviewed combinatorics journal, and several of the integer sequences discovered in that paper are now published on Sloane's Online Encyclopedia of Integer Sequences. Dr. Sloane read the pre-print and sent us a nice email about it!

__List of papers that will be submitted this year:__

1. Dr. Pamela Harris, Darleen Perez-Lavin (an FGCU graduate student), Alexander Diaz-Lopez (a graduate student in his final year at the University of Notre Dame), and I will submit an article called *Peak Sets of Classical Coxeter Groups* to a peer-reviewed journal within the next week.

**Here is the abstract for the paper: **

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.

2. Dr. Pamela Harris and Dr. Mohamed Omar and I are currently collaborating on a paper called *On the value of the q-analog of Kostant’s partition formula on the highest root. *We met at the US Military Academy in February to work on the project, and have made some wonderful progress on the problem in types C. Our summer goal is to finish the paper and solve the problem in types B and D.

I am hopeful that most or all of these articles will proceed through the review process relatively quickly and be publish within the next year or so. Fingers crossed!

__List of talks given this year:__

- Post-Doctoral Research Seminar in Algebra and Topology- Title:
*Studying peak sets of classical Coxeter groups*-- US Military Academy at Westpoint (April 2014) - Joint Mathematics Meetings - AMS Contributed Paper Sessions - Title:
*On k-distance domination of grids*San Antonio, Texas(January 2015)