Chi-Kwong Li, Department of Mathematics

Research Advising

You may vist my publication list 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.

Graduate advising

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 arrangments. 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 preserver problems. [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

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.

REU Projects

• 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)

• 1994: Ann-Louis Klaus (High honors), A study of the vector (p,q) norm 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 values of matrices.
• 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. Kincaid.]
• 2003: H.Chiang (Highest honors), Doubly stochastic matrices and linear preservers.
• 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 control.
• Study the higher rank numerical range associated with quantum error correction.
• 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 [73]).
• Study the polynomial hull of (special classes of) matrices.

• Determine all possible inertia values, ranks, eigenvalues, determinants, etc. of the sum of k matrices with k > 2.
• Determine 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 dimensional operators.
• 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.

• Characterize all possible isometry groups of a permutation invariant norm on C^n (see my 1995 paper [80] and papers [87]).
• 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.

• Determine the maximum determinant of a zero one matrix with fixed line sum (see my papers [85] and [121]).
• 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 Chiang.

• 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 on M_n.

• 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.