As a student of Nick Varopoulos in the late sixties, my first research interests were in the area of Abstract Harmonic Analysis. In 1970, I solved the union problem for Sidon sets which had been open for 35 years. The same idea was applied to a problem relating to Fatou-Zygmund sets in the mid seventies. As time went on I became more and more interested in the underlying inequalities that one finds in analysis rather than abstract concepts and ideas.
In the late seventies and eighties, I worked mainly on problems in Euclidean Harmonic Analysis, spending most of my time on Radon transforms and on estimates for the restrictions of Fourier transforms to curves and surfaces. Perhaps the nicest result proved around this time was the restriction estimate for a class of curves in Euclidean space. This involved a bootstrap method that allowed the range of Lp spaces for which the result is valid to be extended inductively.
In the late eighties and nineties I have started to work on Matrix Theory. I am particularly interested in the application of harmonic analysis to matrix theory, mainly through the central role played by the permutation group. Other interests include graph theory and combinatorics especially in connection with matrices.
More recently I have returned to questions similar to von Neumann's Inequality relating to the symbolic calculus of operators on complex Hilbert space. This is an interest that stems from the work of Nick Varopoulos back in 1970. My most recent contribution, the teardrop theorem shows that the result of applying an analytic function to a Hilbert space operator with unit numerical radius has its numerical range in a teardrop shaped region in the complex plane. The theorem generalizes one proved by Berger and Stampfli some 40 years earlier.
In 2010, I found a counterexample to a conjecture of Matsaev posed in 1966, essentially the Lp version of the von Neumann inequality. The counterexample was found using "hill climbing" methods implemented in part in the CUDA language and using parallel processing on an NVIDIA graphics card.
In the Summer of 2011, I found a proof of the 1983 conjecture of Brualdi and Li on the Perron root of tournament matrices. I am not an expert in this field and don't plan to pursue it further. There are some notes on ideas that didn't pan out accessible from the navigation panel together with some open problems.
In the late Winter of 2012, I discovered a proof of a question raised by Bhatia and Kittaneh on the geometric-arithmetic mean inequality for positive definite matrices.