Research & Personnel Supervison

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Research Interests:

My main research interests are in the areas of linear algebra, matrix theory, and combinatorics. These interests reside in many areas of mathematics ranging from applied to pure mathematics, and also numerical analysis. I am extremely interested in research problems that have applications or require expertise in overlapping areas of mathematics. I continue to work on problems from combinatorial matrix theory, which, of course, overlap graph theory and linear algebra. Recently, I have been working on a number of different problems, which fall under the headings:

(i) Positivity classes of matrices;

(ii) Algebraic graph theory & Combinatorial Matrix Theory;

You can find a reasonably current list of my publications via the AMS website at the following link.

For a wonderful description on the history of matrices and determinants written by J. J. O'Conner and E.F. Robertson click here.

Positivity classes of matrices.

1. totally nonnegative matrices:
   
2. determinantal inequalities
   
3. eigenvalue location problems
   
4. matrix completion problems
   
5. matrix inequalities

An m-by-n matrix A is called totally nonnegative (resp. totally positive ) if the determinant of every square submatrix (i.e., minor) of A is nonnegative (resp. positive). The class of totally nonnegative matrices have been well studied and arise in a variety of applications including differential equations, statistics, mathematical biology, approximation theory, integral equations and combinatorics and many others.

Combinatorial matrix theory.

1. algebraic connectivity and related topics:
   
2. spectra of graphs
   
3. qualitative matrix theory
   
4. minimum ranks of graphs
   
5. algebraic graph theory

Laplacian matrices of weighted graphs have drawn much attention in recent years, since they arise not only in algebraic graph theory but also in sparse matrix computations and resistive electrical networks.

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Student and Postdoctoral Fellow Supervision

Postdoctoral Fellows & Visiting Scholars:

• Dr. Mahmoud Manjegani (visiting from Isfahan University of Technology), September 2009 -- December 2010.
• Dr. Sasmita Barik (Ph.D. from Indian Institute of Technology Guwahati, 2006), Jan. 2008 - Dec. 2008 (co-supervision with S. Kirkland).
• Dr. Francesco Barioli (Ph.D. from University of Padova, 2001), Oct. 2001 -- Aug. 03.
• Dr. Sukanta Pati (Ph.D. from Indian Statistical Institute, 1999), May 2000 -- April 2001, (co-supervision with S. Kirkland).

Graduate Students:

• Yashar Tavakoli, MSc. to begin May 2010.
• Fatemeh Alinaghipour, Ph.D. in progress (co-supervision with K. Meagher)
• Micahel Cavers, Ph.D. in progress (co-supervision with S. Kirkland).
• Yong Jun Xing, Ph.D. in progress.
• Elizabeth Diaz-Bobadilla, MSc. completed (co-supervision with A. Volodin)
• Xiao Ping Liu, Ph.D. completed

Undergraduate Students:

• Taylor Barrett, Totally Nonnegative minors and their combinatorics (NSERC USRA 2008).
• Ryan Lekivetz, Asymmetric P-matrix completion problems (NSERC USRA 2004).
• Ryan Lekivetz, P-matrix completion problems (NSERC USRA 2003).
• Jeremy Hinks, Hadamard duality (NSERC USRA 2003).
• Dan Scellenberg, Investigations for Totally Nonnegative Matrices (NSERC USRA 2003).
• Chris Leurer & Jeremy Hinks, Integral Laplacian spectrum (co-supervision with S. Kirkland), Summer 2002.
• Nathan Krislock, Determinantal Inequalities for Totally Nonnegative Matrices (NSERC USRA 2000).

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