My research goals for this year were to prepare a manuscript for publication, conduct original research, and foster research opportunities for graduate and undergraduate students. I am pleased to say I met all of these expectations. My first publication Patch Ideals and Peterson Varieties (with Alex Yong of the University of Illinois – Urbana- Champaign) was published in the December 2012 issue of Transformation Groups.
I am currently putting the finishing touches on a manuscript Supercoiling Tangles and Applications to Stick Numbers of 2-Bridge Links with Dr. Rolland Trapp of California State University- San Bernardino (CSUSB). Through the wonders of the internet, you may follow this link to view the draft in its latest form Supercoiled Tangles and Stick Numbers of 2-Bridge Links. I am traveling to San Bernardino on July 16-20 to give a research presentation at the CSUSB Research Experience for Undergraduates program and to polish of this manuscript for submission to a peer-reviewed journal.
A minimal stick representation of a 35 crossing link with just 33 sticks.
I successfully incorporated both graduate and undergraduate students into my research program. I met weekly with Christie Mauretour and DavidBlessing to conduct research on weighted domination numbers of grids. I also met weekly with Tyler Medina to research short presentations of cohomology rings for Schubert varieties. Christie Mauretour’s project started as part of her honors contract in my Discrete Mathematics course. Christie, an undergraduate student, discovered several original results, which she presented at FGCU’s first annual ASPiRE Conference, and at the Underrepresented Students in Topology and Algebra Research Symposium (USTARS) in April 2013. Here is a link to Christie's poster titled Weighted Domination Numbers of Directed Grids.
David Blessing, a first-year graduate student, and I have written a program for the mathematical software package SAGE to compute the weighted domination numbers of grids, and we are currently preparing a manuscript for publication. This manuscript describes the algorithm and provides a construction that we conjecture achieves the weighted domination number for any grid. David presented his research at the USTARS Conference as well. Here is a link to David's talk titled Domination Number of Half Grids.
Christie, David, and I are currently preparing a manuscript for publication. We expect to submit this paper by the end of the summer. Hence the paper will likely be submitted by the end of the Fall semester 2013. Click on this link to view the latest draft of the paper Computing Weighted Domination Numbers of Grids.
Tyler Medina, a second-year graduate student, and I have been searching for a short list of ideal generators for cohomology rings of Schubert varieties. This year we disproved a conjecture from 2010 of Reiner, Woo, and Yong that the number of generators necessary to describe the cohomology ring of Schubert varieties grows exponentially with respect to n. We conjecture that the number of generators is less than 2n. Tyler presented his findings at FGCU’s Research Day 2013.
I also attended the Trends in Undergraduate Research in the Mathematical Sciences (TURMS 2012) conference in October. At this conference I learned various "best practices" for overseeing successful undergraduate research projects and discovered new funding opportunities for undergraduate research.