Chi-Kwong Li, Department of Mathematics
My research specialty is on operator theory, matrix theory,
combinatorial theory, and their applications such as numerical analysis,
mathematical biology, quantum computing, economics, etc.
In addition, I have a wide range of interests in mathematics
including mathematical education.
I am happy to do research with graduate and undergraduate students.
You may vist my
to see my joint papers with graduate and undergraduate students
If the papers do not make sense to you, you are welcome to e-mail me or
make an appointment with me to discuss possible projects.
I can do Ph.D. advsing under the Applied Science
Department at William and Mary, or do joint supervision
of graduate students at other universities under appropriate
Visit the webpage of Applied Science
for application deadlines and procedures.
Here is a list of my former graduate students.
- Nung-Sing Sze (2004, University of Hong Kong), A study of
[Joint supervision with Jor-Ting Chan.]
- Tom Milligan (2003, William and Mary),
On certain sets of matrices: Euclidean squared distance matrices,
ray-nonsingular matrices, and matrices generated by reflections.
Honors projects and undergraduate research projects
I am happy to supervise undergraduate research projects
in various topics in the format of Honors projects and REU projects
during the regular semester or winter break, or summer research projects
supported by NSF REU program at William and Mary or any other summer
research grants such as the Wilson Interdisciplinary Fellowship.
In the past, I have conducted undergraduate research projects on matrix
theory, coding theory, game theory, combinatorial theory. Here is a list
of my former students.
- 1990: S.Chang, C.Heckman, (3 projects)
- 1992: P.Mehta, (3 projects)
- 1993: C.Hamilton-Jester, A-L. Klaus, W.Whitney, (3 projects)
- 1996: T.Coleman, J.Lin, T.Travison, (2 projects)
- 2000: H.Chiang, W.Funk, (2 projects during acad. year)
- 2002: F.Curtis, D.Pragel, O.Shenker, K.Spurrier, (2 projects)
- 2003: M.Alwill, E.Bellenot, C.Maher. (2 projects)
- 2005: C.South (1 project)
- 2007: S.Clark, A. Rastogi (2 projects)
- 2008: T. Crowder, J. Mahle, D. Yang (3 projects)
Wilson Interdisciplinary Research Projects
- 1994: Ann-Louis Klaus (High honors), A study of the vector (p,q)
and the induced (p,q) norm.
- 1994: John E. Karro (High honors), An examination of the extreme
points of the set of positive semidefinite doubly stochastic matrices.
- 1997: Carrie Pohanka (honors), Estimating the extreme singular
- 1997: Ingrid Nelson (Highest honors), Error correcting codes on the
towers of Hanoi graphs.
- 1998: Brad Arkin (Highest honors),
Algebraic structures in Feistel ciphers
and an analysis of GOST.
[Joint supervision with W. Bynum.]
- 2003: F.Curtis (Highest honors),
Special Classes of Zero-One Matrices. [Joint supervision with R.
- 2003: H.Chiang (Highest honors), Doubly stochastic matrices and
- 2008: T. Crowder (Higherst honors), A study of genetic code by
combinatorics and linear algebra approaches.
- 1996: Sita Nataraj, A study of game theory.
Some sample research problems.
Numerical ranges and applications
- Determine the C-nuermical radius of A for certain C and A arising
from quantum physics.
- Study the constrained C-numerical range arising from quantum
- Study the higher rank numerical range associated with quantum error
- Find a short proof for the convexity of the intersection
of the quaternion numerical range and the upper half plane; see the work
of R.C. Thompson, Y.H. Au-Yeung, Wasin So, and Fuzhen Zhang.
- Characterize C so that the C-numerical range of A
W(C:A) is always convex (see my 1994 paper ).
- Study the polynomial hull of (special classes of) matrices.
Isometry and preserver problems
- Determine all possible inertia values, ranks, eigenvalues,
determinants, etc. of the sum of k matrices with k > 2.
the best possible containment region for the determinant of the sum of a
real symmetric and a real skew-symmetric matrix.
- Determine the
relations between the singular values and the eigenvalues of the real and
imaginary parts of a matrix.
- Show that intrrinsic unitarily
invariant metric on the Grassmannian must be come from symmetric norms on
the canonical angles.
- Extend matrix inequality results to infinite
- Study the Sendov conjecture.
- Prove that
a norm on M_n induced by a symmetric norm is Schur multiplicative.
Show that a normed vector space V is an inner product space if the norm
attaining vectors for each A in End(V) is a subspace.
Combinatorial matrix theory
- Characterize all possible isometry groups of a
permutation invariant norm on C^n (see my 1995 paper  and
- Characterize all possible isometry groups of an absolute
norm on R^n.
- Characterize the isometry groups of a norm on $M_n$ induced by
a symmetric norm on C^n.
- Suppose $A*B$ is the product A*B = A+B, A-B, AB, ABA, AB+BA, AB-BA,
etc. Determine all the disjointness preserving maps f, i.e., maps on
matrices or operators such that
A*B = 0 if and only if f(A)*f(B) = 0.
- Given a function $F$ on matrices such as the norm, numerical
radius, rank, etc., and a binary operator * of matrices. Characterize
maps g such that F(A*B) = F(g(A)*g(B)) for all A, B.
Positive (linear) maps
- Determine the maximum determinant of a zero one matrix with
fixed line sum (see my papers  and ).
- Given an adjacency matrix A determine a diagonal matrix with
minimum trace such that D-A is positive semidefinite.
- Study the partition problem proposed in the paper with Brualdi and
- Study the extreme points of the set of positive maps
on M_n that preserve diagonal entries.
- Study the extreme points of the set of positive maps
- Applications to mathematical biology.
See my papers with Schneider, Schreiber, Kirkland.
- Applications to learning manifolds. See my paper with Li and Ye.
- Study the optimal parameters for SHH iteration methods.
See my papers with Bai and Golub.
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