The MERiT panel on undergraduate research

Mathematics Education Research in Texas (& undergraduate research)

Recently (Friday, Oct 17, 2014) I had the privilege of being on a panel on undergraduate research, at the MERiT conference in Huntsville, Texas.  Other panelists were Tami Cook, director of the EURECA center at Sam Houston and Kathy Horak Smith, math ed faculty member at Tarleton State University.  The discussion was moderated by Dusty Jones, mathematics educator at Sam.  Each of us, including the moderator, described our experiences with undergraduate research.

Dr. Tami Cook describe her enjoyment of directing undergraduate research in biology and described the development of the undergraduate research center (EURECA) here at Sam Houston.  That center has been open for only a year and Dr. Cook only gets a single course release but she has done a remarkable job in just 12 months, obtaining significant funding for student projects and student travel.  The most significant funding has been for the FAST program (Faculty & student teams) which provides up to $10,000 per summer team (a faculty member and three students) to engage in research or creative endeavors.

I described my experience with undergraduate research, mentioning Ronnie Brown's Carpentry Fable and my summer at the National Security Agency's Director's Summer Program (NSA DSP).  In an attempt to briefly engage my audience (without blackboard or other materials), we iterated some numbers through the 3n+1 problem. I described the Collatz conjecture (hailstone problem) as an example of something easy for undergraduates to understand but (very) hard to solve.  (I will attempt to look at some accessible variations of the Collatz problem in a later post.)

Kathy Horak Smith described her summer work with math seniors in secondary education.  She travels from Tarleton State to Texas Christian University each summer to work with grade school children in an ELL (English Language Learners) math program and takes college students with her.  These college students are engaged in research into mathematical learning, as it is done by young children whose first language is not English.  It was clear from Kathy's exposition that this is an exciting opportunity for her college students and they engage in this professional work with enthusiasm!

Dusty Jones briefly described his summer work with an NSF funded team of five students in the SAM REU.  The students, most of them in an education track, analyzed a variety of grade school statistics textbooks and used the Guidelines for Assessment and Instruction in Statistics Education (GAISE) to evaluate the materials in the textbooks.

Advice from panelists

At the end, we were asked a number of questions about our work.  In response to one question, Tami Cook emphasized the importance of laying out the work expectations for undergrad students at the beginning of a program.  Students needs to be clearly told when work begins, that it will be full-time. The ground rules for work should be laid out early.  Yes, there is much flexibility within any research program (NO, we don't have to start at 8!) but many undergraduates do not really understand either full-time work nor what it means to work 40 hours a week on a science project.  Don't soft-pedal the work involved; the really good students are eager to work hard and dive into the research!

In response to another question, Kathy emphasized the importance of high, professional expectations for our students. If we describe professional expectations, many of our students are eager to step up to those expectations!  They will make mistakes, of course, but the value of the undergraduate research program is to introduce students to their profession.  They are eager to become professionals and are ready to quit being students!  Respect that desire!  (Yes, in my experience directing math research, I can ask my students to learn LaTeX, install GAP or Sage, and although some faculty think that we should not expect this of undergrads, by the time I turn around, the students are done and eager to move on to the next task.)

The presentations were well received by conference participants, with a number of questions after the formal end of the panel and during the lunch that followed.  It was an enjoyable experience to discuss undergraduate research in this setting.

Next time: The linear algebra of strongly regular graphs

[Mathematical prerequisites: This article requires no mathematical background.]

Follow Ken W. Smith on Twitter @kenwsmith54

Strongly regular graphs, I

Strongly regular graphs

Strongly regular graphs "stand on the cusp between the random and the highly structured", says combinatorialist Peter Cameron.  The conditions for the existence and the construction of strongly regular graphs provides a rich source of mathematical investigation, with many applications to algebra, linear algebra and combinatorics.

A graph is regular if every vertex has the same degree.  Let's use the letter k for the degree of our regular graph.

A regular graph is strongly regular if the number of walks of length 2 between two different vertices depends only on the adjacency of the two vertices.  We let  lambda  count the number of paths of length 2 between adjacent vertices and set  mu  as the number of paths of length 2 between non-adjacent vertices.

Given adjacent vertices x and y (in the figure below), the parameter  lambda  counts the number of vertices z adjacent to both.

But if x and y are not adjacent (below), mu counts the number of vertices z in this second configuration.

The parameters of a k-regular graph on v vertices are written as a 4-tuple (v, k, lambda, mu).  The graph below, on 9 vertices, with degree 4, is an example of a (9, 4, 1, 2) strongly regular graph (SRG.)  Each edge (adjacent pair) is on exactly 1 triangle, but each nonadjacent pair form the opposite corners of a 4-cycle and so there are mu=2 walks of length two between the nonadjacent vertices.

The Petersen graph, below, is an example of a (10, 3, 0, 1) SRG.

It is possible for mu and lambda to be equal.  Here (below) is the Shrikhande graph, an SRG with parameters (16, 6, 2, 2).  (This graph was found by S. S. Shrikhande.  I was privileged to work for some time at Central Michigan University with Mohan Shrikhande, son of S. S. Shrikhande.  Both father and son live in Mt. Pleasant, Michigan.)

There is (of course) a nice Wikipedia article on SRGs.  The drawing, above, have been copied from that article.

Fundamental existence question & conditions

A fundamental question in the study of strongly regular graphs is simply, "For which parameters (v, k,  lambda, mu) does there exist a SRG?"  The current progress on this question can be summarized in the tables kept by Andries Brouwer in Eindhoven, Netherlands.  

One first step to eliminating false parameter sets is the "First feasibility condition" created by counting ordered pairs of vertices (y, z) in the configuration below.

We fix the vertex x and count vertices y, z where x and y are adjacent, as are y and z, but x is not adjacent to z.  Given x, there are k choices for the vertex y and then k-lambda-1 choices for z.  (There are k vertices adjacent to y but one of those is x and there are another lambda which are adjacent to x.  Throw those out....)  So, given x, there are k(k-lambda-1) configurations like that, above.

On the other hand, given x, we can hunt for vertices z first -- there are v-k-1 choices for z -- and then hunt for y -- there are mu choices for y once we find z and so there are (v-k-1)mu  configurations like that, above.

Since we got two results for the same count, we know they must be equal and so we have the following condition on the parameters (v, k, lambda, mu) of a SRG:

k(k-lambda-1) = (v-k-1)mu

This is the "first feasibility condition" for SRGs.  There are a few more feasibility conditions, involving more subtle arguments.  We will explore two of those in a later post.

Open parameter sets

Meanwhile, just so one realizes that there are LOTS of open questions on strongly regular graphs, we mention a few "existence" questions here.  It would be very nice to either construct or rule out SRGs with the following parameters:

(65, 32, 15, 16)

(69, 20, 7, 5)

(75, 32, 10, 16)

...

(96, 35 10, 14)

(96, 45, 24, 18)

(99, 14, 1, 2)

(100, 33, 8, 12)

...

(120, 35, 10, 10)

...

(144, 52, 16, 20)

...

(160, 54, 18, 18)

(162, 21, 0, 3)

...

Of the many open cases, I have pulled out in the list above, the parameters that I (and others) find most interesting.  Most of these cases are quite hard and the discovery of a single graph in the list above would be a publishable paper!  (Write me!  Did I mention my Erdos number is 2?!)

All strongly regular graphs on 64 or fewer vertices have been found by exhaustive computer search by Ted Spence.  One might note that there are exactly 32548 distinct (nonisomorphic) graphs with parameters (36,15,6,6).

I have worked pretty hard on the parameter set (99,14,1,2).  A doctoral dissertation in computer science by Majid Behbahani (2009, here in pdf) attempts the most obvious constructions for this strongly regular graph and does not find it.  This does not rule out this SRG but makes it clear that a construction will be somewhat unusual.  In the same vein, Makhnev and Nosov describe a search for the graph (162,21,0,3).   I've also looked into (120,35,10,10) and (160,54,18,18); in both cases there is no abelian Cayley graph with these parameters.  (More on that later.)  

A strongly regular graph with parameters (3250,57,0,1) is an example of a Moore graph.  Many of us in algebraic combinatorics have looked for that graph and it still eludes us.

It turns out that the most powerful tools for the study of strongly regular come from linear algebra.  That will be the subject of the next post.

Next time: The MERiT conference & undergraduate research.

[Mathematical prerequisites: This article requires a basic introduction to graph theory.]

Follow Ken W. Smith on Twitter @kenwsmith54

COURI and EURECA, promoting undergraduate research across campuses

COURI

Last Thursday, while visiting professor Art Duval at the University of Texas at El Paso (UTEP), I had the privilege of spending about an hour with Lourdes Echegoyen, UTEP's director of the center for undergraduate research.  The UTEP center, now called COURI (Campus Office of Undergraduate Research Initiatives) has a full time director (Dr. Echegoyen is a member of the Chemistry faculty), full-time staff, and is even now advertising for an associate director!

The COURI center has expanded interest across campus in undergraduate research.  One way undergraduate research is promoted is through a campus-wide zero-credit class that interested students may take prior to pursuing undergraduate research.

The Center and faculty are active in promoting undergraduate research in every discipline but much research interest comes from science faculty.  The faculty and staff involved in COURI have received several federal grants and are encouraging the development of NSF-REU proposals, although most of their emphasis seems to be on year long, in-house programs.  (In the long run, year long, in-house programs are much more beneficial to students than short summer programs.)  The UTEP on-campus programs have even begun receiving applicants from China as Chinese colleges offer to fund student travel and pay UTEP tuition.  The UTEP undergrad research programs have also recently begun engaging in an exchange program with universities in Mexico.

Dr. Echegoyen is a very energetic individual.  I suspect she spends 60 hours a week administrating, organizing, promoting undergraduate research at UTEP.  It was fun to talk to her about undergraduate research!

During our conversation, I wondered out loud about the possibility of exchanging students in some way, maybe recruiting Sam Houston students to UTEP, or vice-versa, to engage in undergraduate research.  Could we set up some type of online math connection between people at our two schools?

EURECA

Sam Houston State University also has a center for the promotion of undergraduate research.  Although only a year old, the center, Enhancing Undergraduate Research and Creative Activities (EURECA), has a director, Dr. Tami Cook (Biology faculty), two graduate assistants, a budget, and is actively promoting undergraduate research across our campus.  Last summer eight student research teams were funded by EURECA, with the aid of college deans.  These Faculty and Student Team (FAST) summer awards pay both students and faculty.  A faculty member leads a team of typically three students, working on a research project for ten weeks of the summer.  The faculty member is paid $4000 and the students are paid $2000.  Although these stipends are about half of the NSF-REU stipends, they are a good start towards promoting undergraduate research!

Students involved in the FAST summer programs at Sam Houston State University often continue to engage in research with their professor after the formal summer project ends.  (Some had already begun research before the summer funding stimulus.)  In the spring following the summer project, the students present their work on campus at the annual Undergraduate Research Symposium sponsored by the Honors College at Sam Houston.

If I may boast for a moment (this is my blog!) -- I chaired the committee that developed EURECA center and so, of course, I am very proud of that program.  Please read more about the recent developments here!

Next time: Strongly regular graphs

[Mathematical prerequisites: This article requires no mathematical background.]

Follow Ken W. Smith on Twitter @kenwsmith54