Archives For undergrad research

A couple months ago, my colleague Jeff Rushall and I co-applied for a Center for Undergraduate Research in Mathematics (CURM) mini-grant to fund a group of undergraduate students to work on an academic-year research project. Jeff and I had both individually applied in the past, but neither of us were successful in our proposals. If you are interested, you can take a look at my previous proposal by going here. Jeff and I are both passionate about undergraduate research and work well together. We decided that a joint application would likely be stronger than two individual proposals. I’m happy to report that we recently found out that our proposal was funded. We’re thrilled!

Here are a few more details. For the upcoming project, we recruited a diverse group of 7 talented undergraduates: Michael Hastings (one of my current research students), Emily White, Hanna Prawzinsky, Alyssa Whittemore, Levi Heath, Brianha Preston, and Nathan Diefenderfer. Students are expected to spend ten hours per week during the academic year working on the research project. In return, each student will earn a $\$3000$ stipend. Money from the grant will also be used to buyout a single course for both me and Jeff. In addition, Jeff and I will team-teach a topics course each semester that will include our research students but will also be open to other interested students. CURM will cover most of the travel expenses for Jeff and I to attend the Faculty Summer Workshop at Brigham Young University (BYU). CURM will also cover most of the travel expenses for the nine of us to attend the Student Research Conference at BYU.

A collaborative research program has many advantages over operating several disconnected projects, as Jeff and I have done in the past. One of our goals is to build a self-sustaining research group. Ideally, this group will consist of students at different stages of their education, each participating for multiple years. The opportunities afforded by a CURM mini-grant will provide a catalyst for our endeavor in several ways. First, the visibility of a student research group with a CURM mini-grant will help our department recruit mathematically inclined students. NAU has many such students, but some are enticed by existing research groups and grants in our science programs. Second, we would like to take advantage of the mentoring and training the CURM program provides for faculty. One weakness in my past projects is getting students to finish writing up their results for publication. I am hopeful that my involvement in CURM will help remedy this. Third, we believe that the stipend money will enable our students to forgo some of their part-time work and instead devote their time to mathematics. Lastly, we want to use the experience as a stepping stone towards obtaining an externally funded REU program.

Next year’s research project involves “prime labelings of graphs,” which is outside my typical research interests. Jeff and I believe that we have found a project that is accessible to undergraduates yet rich enough that we won’t even come close to running out of stuff to do. I’m really looking forward to branching out and exploring something new.

If you are interested, here is the project description that we submitted.

Project Description

This research project is motivated by a conjecture in graph theory, first stated in a 1999 paper by Seoud and Youssef [1], namely:

All unicyclic graphs have prime labelings.

This is a viable choice as a research problem for undergraduates because it is interesting yet accessible, in large part due to the minimal amount of background information required. To wit, a unicyclic graph is a simple graph containing exactly one cycle. An $n$-vertex simple graph $G$ with vertex set $V(G)$ is said to have a prime labeling if there exists a bijection $f: V(G) \to \{1, 2, 3, \ldots, n\}$ such that the labels assigned to adjacent vertices of $G$ are relatively prime.

As discussed in Gallian’s “A Dynamic Survey of Graph Labeling” [2], many families of graphs have prime labelings; the “simpler” types of unicyclic graphs that are known to have prime labelings include cycles, helms, crowns, and tadpoles. The goal of our project will be to discover additional classes of unicyclic graphs with prime labelings, in hopes of bringing the aforementioned conjecture on all unicyclic graphs within reach. The families of graphs we will investigate include, but are not limited to:

  1. double-tailed tadpoles, triple-tailed tadpoles, etc.;
  2. irregular crowns (crowns with paths of different lengths attached to each cycle vertex);
  3. unicyclic graphs with one or more trivalent trees attached to cycle vertices;
  4. unicyclic graphs with one or more complete ternary trees attached to cycle vertices; and
  5. unicyclic graphs with a specified number of non-cycle cubic vertices.

Seoud and Youssef have established necessary and sufficient conditions for some graphs to have prime labelings, but they are somewhat limited in scope. Seoud has also published an upper bound on the chromatic numbers of prime graphs. These and other results may be beneficial to our students as their research project progresses.

Ernst and Rushall have already made some progress on these specific cases. We will use these initial results as a starting point with our team of students. More precisely, the 7 students involved in this project will attend a 3-credit research seminar during both semesters of the 2014-2015 academic year. Ernst and Rushall will team-teach the seminar, but eventually the students will play an equal role in leading discussions, presenting research results, etc.

Our recent experience with Seoud’s Conjecture has indicated that this problem is ripe with potential and highly appropriate as an undergraduate research project. The students can begin productive work in a single afternoon, and yet we anticipate the students producing original results worthy of publication in refereed journals by the end of the academic year. Moreover, there appear to be a virtually unlimited number of families of graphs to investigate, which will hopefully lead to a sustainable research program for undergraduates in future years.

The 7 students we have recruited to work on this research project are mathematics majors in our department. In addition, they all have very good academic records, and have proven themselves to be hard-working, reliable and creative students in previous courses that we have taught, including vector calculus, linear algebra, abstract algebra, foundations, discrete mathematics and number theory. These students regularly attend our weekly departmental undergraduate seminar, so they are familiar with the rigors associated with research and are motivated to investigate deeper problems in mathematics.

It should be noted that several presentation venues (departmental, university-wide, as well as regional conferences) will be exploited to allow our students an opportunity to showcase their efforts during the 2014-2015 academic year.

References

[1] M.A. Seoud and M.Z. Youssef, “On Prime Labeling of Graphs,” Congressus Numerantium, Vol. 141, 1999, pp. 203-215.

[2] J.A. Gallian, “A Dynamic Survey of Graph Labeling,” The Electronic Journal of Combinatorics, Vol. 18, 2011. http://www.combinatorics.org/Surveys/ds6.pdf

Undergraduate Student Poster Session

The Joint Mathematics Meetings took place last week in Baltimore, MD. The JMM is a joint venture between the American Mathematical Society and the Mathematical Association of America. Held each January, the JMM is the largest annual mathematics meeting in the world. Attendance in 2013 was an incredible 6600. I don’t know what the numbers were this year, but my impression was that attendance was down from last year.

As usual, the JMM was a whirlwind of talks, meetings, and socializing with my math family. My original plan was to keep my commitments to a minimum as I’ve been feeling stretched a bit thin lately. After turning down a few invitations to give talks, I agreed to speak as part of panel during a Project NExT session. The title of the panel was “Tried & True Practices for IBL & Active Learning”. Here, IBL is referring to inquiry-based learning. Last year I gave several talks and co-organized workshops about IBL, so I knew preparing for the panel wouldn’t be a huge burden.

The other speakers on the panel were Angie Hodge (University of Nebraska, Omaha) and Anna Davis (Ohio Dominican University). Each of us were given 15 minutes to discuss our approach to IBL and active learning and then we had about 20 minutes for questions. For my portion, I summarized my implementation of IBL in my proof-based courses followed by an overview of my IBL-lite version of calculus. Angie discussed in detail the way in which UNO is successfully adopting IBL in some of their calculus sections. Anna spent some time telling us about an exciting project called the One-Room Schoolhouse, which attempts to solve the following problem: Small colleges and universities often struggle to offer a variety of upper-level courses due to low enrollment. According to the ORS webpage, the ORS setting allows schools to offer low-enrollment courses every semester at no additional cost to the institution. ORS is made possible by flipping the classroom.

If you are interested in my slides, you can find them below.

I’m really glad that I took the time to speak on the panel. The audience asked some great questions and I had several fantastic follow-up conversations with people the rest of the week.

As a side note, one thing I was reminded of is that speaking early during the conference (which is when the panel occurred) is much better than speaking later. The past couple JMMs, I spoke on the last day or two, and in this case, my talk is always taking up bandwidth in the days leading up to it. It’s nice to just get it out of the way and then be able to concentrate on enjoying the conference.

One highlight of the conference was having a pair of my current undergraduate research students (Michael Hastings and Sarah Salmon) present a poster during the Undergraduate Student Poster Session. Michael and Sarah have been working on an original research project involving diagrammatic representations of Temperley-Lieb algebras. I’ve been extremely impressed with the progress that they have made and it was nice to be able to see them show off their current results. The title of their poster was “A factorization of Temperley–Lieb diagrams” (although this was printed incorrectly in the program). Here is their abstract:

The Temperley-Lieb Algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. Then in 1987, V. Jones showed that the Temperley–Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$ (whose underlying group is the symmetric group). This realization of the Temperley-Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. In the cases when diagrammatic representations are known to exist, it turns out that every diagram can be written as a product of “simple diagrams.” These factorizations correspond precisely to factorizations in the underlying group. Given a diagrammatic representation and a reduced factorization of a group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct the factorization of the corresponding group element. In cases that include Temperley–Lieb algebras of types $A$ and $B$, we have devised an efficient algorithm for obtaining a reduced factorization for a given diagram.

Perhaps I’m biased, but I think their poster looks pretty darn good, too.

All in all, I would say that I had a successful JMM. I had a lot more meetings than I’ve had in the past, so I missed out on several talks I really would have liked to attend. There’s always next year.

SUnMaRC 2013

March 25, 2013 — Leave a comment

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On March 1-3, Jeff Rushall (he’s the Patrick Stewart-looking guy in the photo above) and I, together with a group of 8 Northern Arizona University mathematics majors, attended the 2013 Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at the University of New Mexico in Albuquerque. SUnMaRC is an annual conference that provides undergraduates in the Southwest an opportunity to give 20 minute talks in order to showcase the work they have been doing for their undergraduate research projects. Three of the 8 NAU students are currently conducting research projects under my guidance. That’s them in the picture above (taken during a break at the conference).

Two of my students, Dane Jacobson and Michael Woodward, have spent the spring semester studying the mathematics behind Spinpossible, which is a free game that is available for iOS and Android devices. Alternatively, you can just play the game in any modern web browser. The game is played on a 3 by 3 board of scrambled tiles numbered 1 to 9, each of which may be right-side-up or up-side-down. The objective of the game is to return the board to the standard configuration where tiles are arranged in numerical order and right-side-up. This is accomplished by a sequence of “spins”, each of which rotates a rectangular region of the board by 180 degrees. The goal is to minimize the number of spins used. It turns out that the group generated by the set of allowable spins is identical to the symmetry group of the 9 dimensional hyper-cube (equivalently, a Coxeter group of type $B_9$).

In a 2011 paper, Alex Sutherland and Andrew Sutherland (a father and son team) present a number of interesting results about Spinpossible and list a few open problems. You can find the paper here. As a side note, Alex is one of the developers of the game and his father, Andrew, is a mathematics professor at MIT. Using brute-force, the Sutherlands verified that every scrambled board can be solved in at most 9 moves. One of the goals of my students’ research project is to find a short proof of this fact. During their talk at SUnMaRC, Dane and Michael presented their current progress on this unexpectedly difficult problem. Here are their slides.

A week after SUnMaRC, Dane and Michael gave a similar and improved talk (which includes some random images of Samuel L. Jackson) during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for the FAMUS talks are undergraduates. You can find the slides for their FAMUS talk here.

My third student, Selina Gilbertson, has spent the spring semester working on extending the results of some of my previous students. During the 2011-2012 academic year, I mentored Ryan Cross, Katie Hills-Kimball, and Christie Quaranta at Plymouth State University on an original research project aimed at exploring the “T-avoiding” elements in Coxeter groups of type $F$. Coxeter groups can be thought of as generalized reflections groups. In particular, a Coxeter group is generated by a set of elements of order two. Every element of a Coxeter group can be written as an expression in the generators, and if the number of generators in an expression is minimal, we say that the expression is reduced. We say that an element $w$ of a Coxeter group is T-avoiding if $w$ does not have a reduced expression beginning or ending with a pair of non-commuting generators. My PSU students successfully classified the T-avoiding elements in the infinite Coxeter group of type $F_5$, as well as the finite Coxeter group of type $F_4$. At the time, we had conjectured that our classification holds more generally for arbitrary $F_n$. However, Selina has recently discovered that this is far from true. In Selina’s SUnMaRC talk, she relayed her current progress on classifying the T-avoiding elements in type $F_n$. Here are her slides.

Both sets of students gave wonderful talks. In fact, all 8 NAU students gave fantastic talks. I was impressed. Below are some photos from our trip.