# Research 2013-2014

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 four 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.

__Summary of student research outcomes:__

A number of the student research collaborations I have worked on over the past year and a half 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: Four of my student collaborators presented at our second annual Advancing Student Participation in Research Experiences (ASPiRE) conference, two of them presented at the Joint Mathematics Meetings in Baltimore, three presented at the Embry-Riddle Undergraduate Mathematics Conference, one of them presented at Southeastern International Conference on Combinatorics, Graph Theory, and Computing at Florida Atlantic University, and five of them will present at the Underrepresented Students in Topology and Algebra Research Symposium (USTARS) conference at UC-Berkeley this April. In addition, David Blessing was accepted into the Ph.D. program at Florida Atlantic University, Christie Mauretour was accepted into this summer's MAXIMA Research Experience for Undergraduates (REU) at the University of Minnesota - Twin Cities, and Armando Grez was accepted to three different (REU) programs, and he chose to attend the REU at UT-Tyler this summer.

__List of papers submitted this year:__

I travelled to San Bernardino on July 16-20 2013 to give a research presentation at the CSUSB Research Experience for Undergraduates program and to polish off manuscript I am currently putting the finishing touches on a manuscript on *Supercoiling Tangles and Applications to Stick Numbers of 2-Bridge Links* with Dr. Rolland Trapp of California State University- San Bernardino (CSUSB). Here is a link to the article on arXiv.org *Supercoiled Tangles and Stick Numbers of 2-Bridge Links.* The manuscript is still under review at a peer-reviewed journal.

Dr. Katie Johnson, David Blessing, Christie Mauretour, and I have finished an article on *On (t,r) Broadcast Domination of Grids** *in December 2013. The article is currently under review at a peer-review journal, and I presented the preliminary results at the University of Washington's Combinatorics Seminar. In addition, David Blessing presented the results at the Joint Mathematics Meetings in Baltimore, MD in January, and the Southeastern International Conference on Combinatorics, Graph Theory, and Computing.

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** *in December 2013. 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!

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 earlier this spring. The article is still under review.

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:
*Supercoiled Tangles and Stick Numbers of 2-Bridge Links.*Baltimore, Maryland (January 2014) - UW Combinatorics Seminar - Title:
*On (t,r) broadcast domination numbers of grids*-- University of Washington (October 2013) - CSUSB REU - Title:
*Supercoiled tangles*-- California State University at San Bernardino. (July 2013)