Research Experiences with Undergraduates


Chi-Kwong Li

Department of Mathematics

College of William & Mary

Williamsburg, VA 23187-8795





            When first approached to write an article for IMAGE about the REU program at William and Mary, I wasn't sure there was anything new for me to say, as the paper [JL] already clearly described the program. But then, Hurricane Isabel hit Virginia. The College was closed and computer systems were down. This gave me some more time to think about the project. I came up with the idea to focus the article on some of my personal experience in doing research with undergraduates. In the last 14 years, I have worked with 24 undergraduate students on a number of research projects; see the reference list. For whatever it is worth, here is my story.


What types of undergraduate research programs have I participated in?


            I have participated in several different types of undergraduate research programs including: National Science Foundation (NSF) Research Experiences for Undergraduate (REU) programs conducted in the summer, NSF supplementary REU programs conducted during the academic year, Honors projects for mathematics majors and the Wilson Interdisciplinary research program at William and Mary. Accordingly, I selected students or was selected by students in a variety of ways.


For each of the summer NSF REU programs, eight to nine students were recruited from different institutions. In the first two days of the eight-week program, several potential advisors would present their research projects. Students would then have a meeting among themselves to determine a matching between advisors and advisees. It is amazing that it always worked well with students spread rather evenly among the advisors.


            The NSF supplementary REU opportunities were limited to William and Mary students. Sometimes I invited outstanding students who were taking my courses to participate, and other times I offered the vacancies to good students who inquired about possible research opportunities. The latter approach is the standard way to get students for Honors projects and other research programs at our College.  Knowing that I am interested in advising Honors projects and other undergraduate research projects, students would talk to me about such possibilities. Usually, they were encouraged to talk to other potential advisors as well. In any event, I did get a number of good students working with me in this way.



What kind of research have I done with students?


            It is not hard for readers, especially for those who know me, to guess the answer: matrix analysis! Instead of boring the readers with the technical details of various students' projects, I will only touch upon some of them later when I discuss why I think that matrix analysis is a good theme for undergraduate research.  Here let me mention the few exceptional cases, that is, those research projects with undergraduates with topics other than matrix analysis.


In [LN], a student and I studied coding theory related to the familiar Tower of Hanoi puzzle. This was actually an extension of the student's summer REU project at another university. When I filled out the recommendation form of the other university’s REU program for the student, one of the questions was whether a faculty member at the student’s home university would continue to work with the student after the summer if the student would be interested in doing so.  I said yes to the question and the student was admitted to the REU program. After she came back to William and Mary, she expressed interest in continuing the research.  So, I kept my promise, and worked with her in the following academic year. The research led to [LN], which contains a short proof of the result and the answer to an open problem posed in a paper of the student and her REU advisor.


In the spring of 1997, I taught a course in applied abstract algebra covering topics including some coding theory and cryptology. A student in my class was a double mathematics and computer science major. The student was concurrently enrolled in a computer science class concerning the implementation of crypto systems. He was very interested in both the theoretical and practical aspects of cryptology, and ended up doing an Honors project on cryptology under the joint supervision of a colleague in the computer science department and me. When he graduated, he was hired by a software security company—of course, with a salary much higher than mine. He later learned that he was selected over many applicants with Masters degrees because of his course work and research in cryptology. Two years later, he and his colleagues made CNN news for cracking an online casino by showing that the pseudo-random number generator used to deal the poker game was very insecure. They illustrated how one could predict the poker hands after observing the game for an hour or so. This remains  one of my favorite stories for my abstract algebra students who do not find abstract algebra interesting and useful!


            The next case is just half exceptional because the story started with chaos and ended in matrices. In the spring of 1996, an economics student approached me about the possibility of doing a Wilson interdisciplinary research project in the following summer. Based on a Time magazine article that had piqued her interest, she wanted to work on chaos theory and economics. I frankly told her that I knew nothing about chaos, but I was willing to learn some chaos with her in addition to learning some economics theory from her! She obviously realized that it would be too heavy a burden for both of us. So, she looked into other possibilities, and found another article about game theory and auctions in Forbes magazine. When she asked me about that, I told her that at least I knew matrix games.  So, we ended up doing a project in game theory and economics. In fact, I was so fond of the subject that I taught a topics course in game theory in the following semester in which I discussed some applications of game theory in biology. Some results obtained in that summer and the following semester led to [LNa].  It was quite an educational experience for the student as well as myself.



Why is matrix analysis a good theme for undergraduate research?


            In my opinion, matrix analysis is an excellent topic for undergraduate research. It does not require a lot of background to understand some research questions, yet it is linked to different topics such as group theory, operator theory, operator algebras, and numerical analysis, and it offers endless opportunities for further research.  In fact, the many different aspects of matrix analysis can attract students with different backgrounds. In my work, for students with strong abstract algebra background, we studied homomorphisms or linear/additive maps that leave invariant symmetric groups, alternating groups, semi-groups of stochastic matrices, and other related nonnegative matrix sets [AM,ChL1,ChL2]; for students interested in complex analysis and functional analysis, we studied numerical range [LMR2,LSS] or isometry problems [CL2,Cet,KL,LM];  for students interested in combinatorics, we studied topics in combinatorial matrix theory [CP,LLR,SS]; for students interested in convex analysis, we studied geometrical structure of matrix sets [HL];  and for students with a computer science background, we used scientific computation to study matrix problems [CL1,CP,He]. In fact, advising undergraduate research projects in matrix analysis well manifests the theory of Confucius that "students should be educated and trained according to their strength". (This is truly from Confucius and not from a fortune cookie!) 


            According to the nature of the research problems, students may need to use or develop techniques in group theory, combinatorial theory, functional analysis or scientific computation, in the matrix analysis research projects. This exposed students to different research areas in addition to matrix analysis, and might influence their future choices of research topics in graduate studies. Moreover, the techniques acquired in the projects might be useful in their future research in mathematics or other subjects. For example, the matrix techniques developed in [LNa] were later used in the graduate study in economics by the student (see [Na]).


What have students and I gained by doing undergraduate research projects?


            Students received stipends for their summer research, and Honors project students graduated with honors. Students acquired some experience in mathematical research and got a glimpse of how professional mathematicians work.  In some cases, the research led to the excitement of their first publication.  In any event, students at least learned some mathematics that might be useful for their future study.  On the one hand, I am glad to see that most of my undergraduate research students have gone on to graduate school to study mathematics and related subjects. On the other hand, as long as the students have seen a real picture of what mathematical research is about, I do not have any problem of seeing them pursue directions other than mathematics.


I received some stipend for doing the summer REU projects. Other projects had no financial compensation. Nevertheless, successful research projects led to research papers, a better CV for tenure, promotion, and even for faculty award nominations. Similar to my other research projects, it was most enjoyable to develop with collaborators new ideas to solve problems. Moreover, I have acquired a lot of knowledge through studying new topics with students or through consultation with colleagues on problems arising in the research. All of these are good. But there is a primitive motivation for me to do research with undergraduates.  Researchers, educators, and grant agencies may emphasize that undergraduate experiences can help train young scientists. In comparison, I have a more elementary goal: to let more young people know what mathematics research is about.


I like mathematics, I like mathematics research, I like to share my research experiences with others, and I feel that appreciating mathematics should not be restricted to a small group of people. Not everyone has to be a musician, but many people can appreciate good music. Similarly, I would like to see that more people can appreciate mathematics and mathematical research work—though not every one has to be a research mathematician!


In the following reference list, names of undergraduate students are in italics.


[AM]     M. Alwill and C. Maher, Multiplicative Maps on Matrices, REU report (Advisors: C.K. Li and N.S.   

  Sze), William and Mary, 2003.


[A]         B. Arkin, Algebraic structures in Feistel Ciphers and an analysis of GOST, Honors Thesis (Advisors:  

  C.K. Li and W. Bynum), William and Mary, 1998.


[B]         E. Bellenot, Effects of Biological Invasions on Ecological Communities, REU report (Advisors: C.K.

  Li and S. Schreiber), William and Mary, 2003.


[Bet]      V. Bolotnikov, C.K. Li, P. Meade, C. Mehl, and L. Rodman, Shells of matrices in indefinite inner  

  product spaces, Electronic Linear Algebra 9 (2002), 67-92.


[CL1]     S. Chang and C.K. Li, A special linear operator on M_4R), Linear and Multilinear Algebra 30 (1991), 

  65-75, (based on an REU project).


[CL2]     S. Chang and C.K. Li, Certain isometries on R^n, Linear Algebra and Appl. 165 (1992), 251-261,

  (based on an REU project).


[ChL1]  H. Chiang and C.K. Li, Linear maps leaving the alternating group invariant, Linear Algebra Appl. 340

  (2002), 69-80, (based on an REU project).


[ChL2]  H. Chiang and C.K. Li, Linear maps leaving invariant subsets of nonnegative symmetric matrices,

  Bulletin of Australian Math. Soc., 10 pages, (based on an Honors thesis).


[Cet]     T. Coleman, C.K. Li, M. Lundquist, and T. Travison, Isometries for the induced c-norm on square

 matrices and some related results, Linear Algebra Appl. 271 (1997), 235-256, (based on an REU   



[CP]      C. Curtis, J. Drew ,C.K. Li, and D. Pragel, Central groupoids, central digraphs, and zero-one matrices

             A satisfying  A^2= J, J. of Combinatorial Theory, Series A , 15 pages, (based on an REU report).


 [HJ]    C. Hamilton-Jester and C.K. Li, Extreme vectors of doubly nonnegative matrices, Rocky Mountain J. 

              of Math. 26 (1996), 1371-1383, (based on an REU project).


[He]     C. Heckman, Computer Generation of Nonconvex Generalized Numerical Ranges, REU report 

              (Advisor: C.K. Li), William and Mary, 1990.


[JL]        C.R. Johnson and D. J. Lutzer, A decade of REU at William and Mary,  of the Conference on Summer 

  Undergraduate Mathematics Research Programs 19-29, American Mathematical Society, 2000.


[KaL]     J. Karro and C.K. Li, A unified elementary approach to matrix canonical form theorem, SIAM Review 

  39 (1997), 305-309, (based on an Honors thesis).


[KL]      A.-L. S. Klaus and C.K. Li, Isometries for the vector (p,q) norm and the induced (p,q) norm, Linear and 

              Multilinear Algebra 38 (1995), 315-332,(based on an Honors thesis).


[LLR]     C.K. Li, J. Lin, and L. Rodman, Determinants of Certain Classes of Zero-One Matrices with Equal  

              Line Sums, Rocky Mountain J. of Math. 29 (1999), 1363-1385, (based on an REU project).


[LM]      C.K. Li and P. Mehta, Permutation invariant norms, Linear Algebra Appl. 219 (1995), 93-110, (based 

              on an REU project).


[LMR1] C.K. Li, P. Mehta, and L. Rodman, Linear operators preserving the inner and outer c-spectral, Linear

  and Multilinear Algebra 36 (1994), 195-204.


[LMR2] C.K. Li, P. Mehta, and L. Rodman, A generalized numerical range: The range of a constrained

  sesquilinear form, Linear and Multilinear Algebra 37 (1994), 25-50, (based on an REU project).


[LNa]     C.K. Li and S. Nataraj, Some matrix techniques in game theory, Mathematical Inequalities and

  Applications 3 (2000),133-141,(based on a Wilson interdisciplinary research project).


[LN]      C.K. Li and I. Nelson, Perfect Codes on the Towers of Hanoi Graph, Bulletin of the Australian Math.  

  Soc. 57 (1998), no. 3, 367-376, (based on an Honors thesis).


[LP]     C.K. Li and C. Pohanka, Estimating the Extreme Singular Values of Matrices,Mathematical Inequalities 

 and Applications 1(1998), 153-169,(based on an Honors thesis).


[LSS]    C.K. Li, S. Shukla, and I. Spitkovsky, Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on hermitian pencils, Linear Algebra Appl. 270 (1997),323-349, (based on an REU project).


[LW]      C.K. Li and W. Whitney, Symmetric overgroups of S_n in O_n, Canad. Math. Bulletin 39 (1996), 83-

              94, (based on an REU project).


[Na]       S. Nataraj, Age Bias in Fiscal Policy: Why Does the Political Process Favor the Elderly?, Ph.D. thesis,  

              Stanford University, 2002.


[SS]        O. Shenker and K. G. Spurrier, Notes on ray-nonsingularity, REU report (Advisors: C.K. Li and T. 

               Milligan), William and Mary, 2003.