Dr. Wayne Barrett, Department of Mathematics

Research Results/Findings

This MEG grant made it possible to support several student research activ- ities from Spring 2009 through Winter 2010.

• It funded one of four undergraduate students that I mentored sum- mer 2009 in the BYU Mathematics Department’s National Science Foundation-sponsored Research Experiences for Undergraduates (REU) in accordance with BYU’s agreement with the National Science Foundation to provide partial support for student wages administered under this grant. The four students were Camille Jepsen – Brigham Young University, Robert Lang – Florida Atlantic University, Emily McHenry – Xavier University, and Curtis Nelson – Brigham Young University. One outcome of this 7 week program was the submission for publication of a 30 page manuscript co-authored by the five of us and my Masters student, Kayla Owens. Immediately following the conclusion of the REU program, the four students I mentored attended the sum- mer meeting of the Mathematical Association of America in Portland, Oregon. Camille and Emily presented a joint talk on their work in the student section of the conference as did Robert and Curtis. Both pairs received an Outstanding Presentation Award for the talks they pre- pared while they were at BYU. Moreover, the paper was published in January 2010 in the Electronic Journal of Linear Algebra [1], one of the leading journals in the field. This paper essentially solved the fourth proposed problem in the proposal by developing several new techniques which have continued to be applicable in newer research. I could not have been more pleased with this successful outcome.
• In Spring 2009 term the MEG grant funded Kayla Owens who used this support to both complete her Masters degree and to mentor four of my undergraduate students during a 5 week period that I was gone from the country. One of these was Curtis Nelson who was one of the REU students mentioned above. He worked almost full time during this period supported by Mathematics Department Undergraduate Mentoring funds and a large part of his research contributed directly to the paper [1] mentioned above. Curtis says his progress during those weeks would not have been possible without Kayla’s mentoring.
• In Summer term the MEG funded Mark Kempton, my other Masters student. He conducted research for his thesis and mentored two of my undergraduate students, Seth Gibelyou and Ted Law. I had limited time to mentor them during this period because of my commitment to be with the REU students almost full time. Seth wrote a substantial computer program that calculated a graph theoretic parameter related to the minimum rank problem. Ted found a much simpler proof of a theorem due to my colleague, Raphael Loewy, at the Israel Institute of Technology and one of his Masters students, Efrat Bank. It was not sufficient to publish without additional results, but continues to be most illuminating.
• The mentoring funds were used in Fall 2009 to support my Ph.D. student, John Sinkovic. John completed a Masters degree in mathematics with me at BYU in December 2006 and in Fall 2007 became a Ph.D. student at Eindhoven University, the Netherlands, working with my colleague, Hein van der Holst. John made excellent progress in Eindhoven and his first paper [2] was published in Linear Algebra and its Applications, the primary journal in the field, in April this year. This paper has already been cited in another published paper and has been cited in a number of submitted papers. However, John’s oldest son, who was born with spina bifida, needed an intricate operation that is not available in the Netherlands, so the family moved back to Utah for the operation this August at Primary Children’s. John was too late to receive funding to begin a Ph.D. pro- gram at BYU, but encouraged by our Graduate Coordinator and Department Chair, I was able to use MEG funds to support him for Fall 2009. This aid was of course very timely for John and his family, but had the added advantage that he has been of great assistance this past year and two months in mentoring a team consisting of my Masters student, Mark Kempton, and four undergraduate students. I could not have managed without his help because of heavy departmental responsibilities that fall. There are no other faculty members in the Department who can assist me in mentoring in this research area, but John is more like a fellow faculty member than a student. John and Mark met with 5 undergraduate students throughout fall semester, and in addition, Mark and John spent many hours talking about another problem on their own. The eventual outcome of these interactions was two submitted papers in summer 2010 [3], [4]. It is more appropriate to report on these papers in the following interim report on the MEG grant awarded to me December 30, 2009.

With the MEG funds I have been able to leverage funds provided by the Mathematics Department and National Science Foundation for undergraduate mentoring and bring in graduate students as mentors. This was much more effective and successful than if I had attempted all of the mentoring on my own.

Academic Objectives

The objectives of this proposal were met for many students. Kayla Owens complete a Masters degree in July 2009 and from May to August 2009 was an effective mentor for 7 undergraduates students. Four undergraduates associated with the 2009 REU became first time authors. Curtis Nelson completed a bachelor’s degree in April 2010, has now co-authored a second paper, and been accepted into the Masters program in mathematics at BYU. Emily McHenry completed a bachelors in mathematics at Xavier University and received an excellent graduate fellowship in Mathematics at Louisiana State University. Camille Jepsen and Robert Lang became very young first time authors. Camille is now serving a mission in South Carolina while Robert is completing undergraduate degree in Florida. Seth Gibelyou repeatedly wrote significant programs related to the research and provided many wonderful mathematical insights. He graduated in August in Electrical Engineering and has begun work outside of BYU so we no longer have him as a collaborator. Nicole Thiesemann’s work in the mentoring environment during Fall 2009 has enabled her to become a co-author on a submitted paper. Mark Kempton completed his Masters degree in June 2010 and has begun a Ph.D. program in Mathematics at the University of California, San Diego, one of the top 20 graduate mathematics programs in the United States.

Mentoring Environment

Mathematics research has historically not been accessible to undergraduates but that is changing. With the assistance of mentoring of outstanding graduate students with the support of MEG funds, I have been able to bring quite a number of exceptional undergraduate students into contributing to first class research. Having one or two graduate students help several undergraduate students, as enumerated above, has been an ideal set-up.

Students participating and academic deliverables

Kayla Owens, Curtis Nelson, Seth Gibelyou, Ted Law, Mark Kempton, Camille Jepsen, Robert Lang, Emily McHenry, John Sinkovic, Nicole Thiese- mann. Four of these were supported by MEG funds, and the remaining by College and Department funds.
One published paper and preliminary work for two additional submitted papers, and two award winning talks at the Summer meeting of the Mathe- matical Association of America in Portland, 2009.
John Sinkovic and Mark Kempton each gave an invited talk in April 2010 at sectional meeting of the American Mathematical Society in St. Paul, Minnesota on their very significant, in my opinion, paper [3] below.

Allocation of funds

Kayla Owens	$2,500
Curtis Nelson (REU)	$3,100
Mark Kempton	$2,500
John Sinkovic	$5,000
Travel expenses	$900
Total	$14,000

Of the travel expenses, $800 were applied to the travel expenses of John Sinkovic and Mark Kempton to speak at the American Mathematical Soci- ety meeting mentioned above. An additional $412 came from a second MEG grant. $100 was allocated to travel associated with the REU. I note that $3,000 was requested for travel in this MEG grant. I explained to Professor Dorff who coordiinated the REU for BYU that I would contribute $ 3,000 toward travel expenses for the students at the conclusion of the REU. However, it was simpler for him to use the MEG funds I had directed toward the REU for student wages. In return, the National Science Foundation paid the travel costs for my students to the Portland meeting, so I was satisfied that the original intent of the funds requested for travel were satisfied.

[1] Wayne Barrett, Camille Jepsen, Robert Lang, Emily McHenry, Curtis Nelson, Kayla Owens, Inertia sets for graphs on six or fewer vertices, Electronic Journal of Linear Algebra, 20 (2010) 53 – 78. http://www.math. technion.ac.il/iic/ela/ela-articles/articles/vol20_pp53-78.pdf
[2] John Sinkovic, Maximum nullity of outerplanar graphs and the path cover number, Linear Algebra and its Applications, 432 (2010) 2052 – 2060. [3] Mark Kempton and John Sinkovic, Minimum rank of outerplanar graphs, submitted to Linear Algebra and its Applications, July 2010. [4] Wayne Barrett, Seth Gibelyou, Mark Kempton, Curtis Nelson, William Sexton, John Sinkovic, and Nicole Thiesemann, The inverse eigenvalue and inertia problems for minimum rank two graphs, submitted to Electronic Journal of Linear Algebra, July 2010.