NEW HOT PAPERS - 2008

 Why do you think your paper is highly cited?

I was able to capture a large literature at a time at which exciting new developments are being made. The paper was written in homage to my long-term collaborator Simon Fitzpatrick (1953-2004) who lost a 10-year battle with cancer in 2004.

In 1988, Simon introduced a new convex function F_A—nowadays called the Fitzpatrick function, which was largely overlooked until the last decade—associated with a monotone operator A, and similarly a monotone operator G_f associated with a convex function f.

I set myself the goals of (i) explicating everything one could do with his ideas and (ii) completing the somewhat unformed discussions we had had by email in the months before his death. Over the next 15 months, I more than met my own goals.

 Would you summarize the significance of your paper in layman's terms?

Monotone operators were invented in the early '60s as a way of studying solutions to (elliptic) partial differential equations. They also capture the first-order behavior of convex functions. As such they are quite fundamental objects both analytically and algorithmically in fields such as functional analysis, optimization, mathematical economics, and elsewhere.

After their basic theory was worked out in the sixties, the remaining questions seemed inaccessible until researchers such as Jean-Paul Penot, Stephen Simons, Nami Fux Svaiter, Mircea D. Voisei, Constantin Zalinescu, and many others began to insightfully exploit the properties of the Fitzpatrick function (as noted this is a convex function which captures much of the structure of these more general monotone objects).

 How did you become involved in this research, and were there any problems along the way?

I wrote my MSc thesis in Oxford on the subject in the early 70s and have continued to stay current with the subject since then. From roughly 1975 to 1990 there was very little progress because we had no tools.

 Where do you see your research leading in the future?

Month by month, there are new connections being found and significant new results being made public. It is a very exciting time for the subject. I would expect my research and that of my colleagues to resolve the major questions—open for many decades—within the next three to five years.

 Do you foresee any social or political implications for your research?

Only that it will keep a few more rowdy mathematicians off the streets while they work on these ideas.

Jonathan Borwein, FRSC
Canada Research Chair in Collaborative Technology
Dalhousie University
Halifax, NS, Canada
2008 Visiting Professor Laureate on Sabbatical at University of Newcastle NSW

Jonathan M. Borwein is featured in ISIHighlyCited.com

Keywords: maximal monotonicity, convex analysis, Simon Fitzpatrick, the Fitzpatrick function, monotone operators, (elliptic) partial differential equations, functional analysis, optimization, mathematical economics, general monotone objects.