**Research Experiences with
Undergraduates**

**__________________________________________________________________________**

**Chi-Kwong
Li**

**Department of Mathematics**

**College**** of ****William**** & Mary**

**Williamsburg****, ****VA**** ****23187-8795**

**Prelude **

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

**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,

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

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

**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

project).

[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

[LN]
C.K. Li and *I.** Nelson*,
Perfect Codes on the Towers of

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

[LP] C.K. Li and *C*. *Pohank*a,
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,

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

Milligan),
William and Mary, 2003.