Archives For 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.


[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.

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.

Several weeks ago, links to a survey article by Jeffrey Lagarias about Euler’s work and its modern developments and a blog post by Richard J. Lipton that discusses Lagarias’ paper were circulated on Google+. I’d like to thank Luiz Guzman and Joerg Fliege for first bringing these items to my attention.

Lagarias’ paper is full of lots of yummy goodies, but my favorite part is his summary of Euler’s approach to research (see Section 2.6 of the paper or the end of Lipton’s post).

Euler’s Research Rules

Taken directly from the paper, here is Lagarias’ summary of Euler’s research rules.

  1. Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
  2. Read and digest every earlier attempts at a theory of the phenomenon in question.
  3. Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
  4. If two special problems solved seem cognate to each other, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
  5. Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
  6. Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.

Lipton’s blog post also lists “Euler’s Research Rules.” My main motivation for reposting them here is to remind myself to follow them!

On Tuesday, February 5 (my birthday!), I gave a talk titled “A diagrammatic representation of the Temperley-Lieb algebra” in the NAU Department of Mathematics and Statistics Colloquium. Here is the abstract.

One aspect of my research involves trying to prove that certain associative algebras can be faithfully represented using “diagrams.” These diagrammatic representations are not only nice to look at, but they also help us recognize things about the original algebra that we may not otherwise have noticed. In this talk, we will introduce the diagram calculus for the Temperley-Lieb algebra of type $A$. This algebra, invented by Temperley and Lieb in 1971, is a certain finite dimensional associative algebra that arose in the context of statistical mechanics in physics. We will show that this algebra has dimension equal to the nth Catalan number and discuss its relationship to the symmetric group. If time permits, we will also briefly discuss the diagrammatic representation of the Temperley-Lieb algebra of type affine $C$.

And here are the slides.

Despite the fact that this was a 50-minute talk, it was intended to be an overview of one aspect of a long and complex story. The subject matter is intimately related to my PhD thesis, as well as a series of papers that I have written.

  • Ernst, D. C. (2010). Non-cancellable elements in type affine $C$ Coxeter groups. Int. Electron. J. Algebra, 8, 191–218. [arXiv]
  • Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley-Lieb algebra, I. J. Pure Appl. Alg. (to appear). [arXiv]
  • Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II. [arXiv]

In addition, there is (at least) a part III that goes with the last two papers that is in progress.

My first semester at Northern Arizona University ended a little over a week ago. Well, it wasn’t my first semester at NAU. I finished my masters at NAU in May of 2000 and then worked as an instructor for the 2000-2001 academic year in the Department of Mathematics and Statistics. Returning to NAU and Flagstaff is a dream come true for me and my family.

What have I been up since I was last at NAU?

After leaving NAU in 2001, I worked for two years as a full-time math faculty at Front Range Community College in Boulder and Longmont, CO. I loved teaching at FRCC and felt like Superman everyday when I went to work. However, I had a hankering to earn my PhD, and after my wife finished her masters, I decided to return to graduate school. I started working on my PhD at University of Colorado at Boulder in August of 2003, and under the guidance of Richard M. Green, I finished in the summer of 2008.

After completing my PhD, I spent four years as an assistant professor at Plymouth State University. PSU is a predominately teaching-oriented institution with low research expectations. Each semester, I taught 3 or 4 different courses. My teaching duties always included one section of Calculus I or Calculus II and my remaining classes usually consisted of upper-level proof-based courses.

While at PSU, I was twice awarded the Plymouth State University Distinguished Professor of Mathematics award, an honor determined by the mathematics majors. Moreover, during my last semester at PSU, I was selected as the university’s sole nominee for the NH Excellence in Education Award, which is a statewide teaching award.

It was during my second year at PSU that I first started using an inquiry-based learning (IBL) approach in my classes. My strategy has been to implement IBL in my proof-based courses first and then work my way down into the calculus sequence. Due to content pressure and class size, using a full-blown IBL approach in calculus is difficult, but not impossible. I would call my current approach in calculus IBL-lite. For me, it is a work in progress.

Despite my high teaching load, I managed to maintain a somewhat active research program. While at PSU, I published three papers and began working on a few more. In addition, I gave frequent math and math education related talks. For the last three years at PSU, I also mentored year-long undergraduate research projects.

Reflection on the past semester at NAU

The teaching load at NAU is much lower, by about half, but the research expectations are much higher. This past semester, I taught two sections of Calculus I. During my last semester at PSU, I taught Calculus II, Calculus III, Linear Algebra, and Logic, Proof, & Axiomatic Systems (which is an introduction to proof course). In addition, I was mentoring three undergraduate research students. You would think that with only teaching two classes with only a single prep, I would feel like I had tons of time to get things like research done! However, I never quite felt that way. I have to cut myself some slack since starting a new job does require some time to transition.


In my view, my teaching went well this past semester. Each of my sections had roughly 45 students, which is way larger than any calculus class I’ve taught before (or care to for that matter). The class met four days per week and for most weeks, one whole class meeting was devoted to students presenting problems on the board. This was well-received by the students and seemed to be beneficial. Unfortunately, I spent most of my time lecturing during the remaining three days per week. The students didn’t seem to mind this approach; it is what they are used to after all. However, I’d like to lecture a lot less.

Even though I’ve taught Calculus I numerous times, it required more effort than I expected to adapt to teaching at a new school with new expectations. Part of the issue is that calculus at NAU is somewhat coordinated and I didn’t want to stand out too much by doing something vastly different than everyone else. This seems overly cautious to me now. I have vague plans for how I can incorporate more IBL and less lecturing, but I still need to flesh out the details.

I held five hours of office hours per week, which were wildly popular. On most days, I had at least three students
for the full hour and it wasn’t uncommon for me to have close to ten. Office hours are my favorite part of my job! My approach is to get the students helping each other. The photos below illustrate a typical day in office hours.

Office Hours

More Office Hours

I also supervised an independent study with one graduate student (Kirsten Davis) on the combinatorics of Coxeter groups. If all goes according to plan, Kirsten will receive approval from the graduate committee and will begin writing a masters thesis under my guidance next semester. I’m excited and nervous about supervising my first graduate student.


As for my research, it has been a mixed bag. I did not manage to get much writing done and did not submit any new articles for publication. However, I have managed to get some writing done now that the semester is over and plan to submit a paper before the spring semester starts. This will make me feel significantly better. If the break was a bit longer, I could get close to having a second paper done. The goal is to find ways to be more productive next semester.

On a more positive note, I did give several talks during the semester. In fact, I gave more talks this semester than I have ever given in a single semester. Here are the details:

In addition to giving talks, I also applied for two grants, both of which are still pending.

During my last year at PSU, I applied for and was awarded a mini-grant from the Academy of Inquiry-Based Learning that provides summer salary to fund the development of IBL course materials for an abstract algebra course that emphasizes visualization and incorporates technology. You can read more about my plan by going here. Since I ended up leaving PSU, I arranged to have the grant postponed until the summer of 2013. So, technically, I’ve already been awarded one grant since arriving at NAU. Woot.

Miscellaneous Highlights

Here are a few other miscellaneous highlights from my first semester at NAU.

  • I’ve got three students lined up to do undergraduate research next semester. The details will get sorted out as we go, but the tentative plan is for one of them to work on a project by herself and for the other two to work together on something different. Regardless, the projects will be in the general area of combinatorics of Coxeter groups. I’m really looking forward to working with these students.
  • This past semester, Tatiana Shubin (San Jose State University) spent her sabbatical on the Navajo Nation. While she was there, she established several Math Circles at middle and high schools. A few weeks ago, Nandor Sieben and I traveled out to the Navajo Nation for a couple days to have Tatiana introduce us to a some of the teachers and administrators that she has been working with, as well as observe some of the circles. Beginning next semester, Nandor and I will be part of a team that will take turns visiting each of the circles in an effort to sustain what Tatiana has started and to support the local teachers that will be running them in her absence. I’m thrilled to be a part of this project.
  • Last summer, Stan Yoshinobu, Angie Hodge, and I organized a contributed paper session at MathFest titled “Inquiry-Based Learning Best Practices.” A few weeks ago, we submitted an abstract to the MAA to organize a similar session and we recently found out that our proposal was accepted.
  • Angie Hodge and I were recently designated as Special Projects Coordinators for the Academy of Inquiry-Based Learning. This position comes with a small annual stipend and our duties include spreading the word about IBL and organizing workshops and conferences like the one mentioned above. I’m extremely passionate about inquiry-based learning and I’m really looking forward to playing an active role in inspiring others to take a more student-centered approach to teaching.
  • I’m a lot like a dog. If I don’t get out for exercise, I might start chewing the furniture. I am more productive at work and a better father and husband if I’m getting regular exercise. Up until some tendonitis in my ankle slowed me down, I was doing a great job of getting outside to run, bike, or climb. In September, I ran my first 50K trail race. Obviously this isn’t academic related, but I’m proud of the fact that I’ve been able to balance work, family, and play enough to train for such an event.


All-in-all, I’d say that I had a pretty good first semester. My one major weakness is not having submitted a paper yet, but I hope to remedy this soon. It’s unfortunate that this may be the only thing that some people care about. I really enjoy doing research, but I’ll never be the world’s strongest researcher. I’m confident that I can do enough to get by (i.e., get tenure at NAU) and I am content being a well-rounded academic.

I work too much. Period. Work is constantly competing for time with family and time for exercising and playing. I feel like this semester was an improvement over recent semesters in terms of balancing work, family, and play. However, I can do better. I need to spend more time with my wife and sons. Of course I want to be the best mathematics professor I can be, but not at the sacrifice of my family. I don’t think about tenure every day, but it’s definitely more on my mind than it used to be at PSU. Having written this reflection, I feel a lot more confident moving forward.