Jonathan M. Borwein talks
with ScienceWatch.com and answers a few questions
about this month's New Hot Paper in the field of
Mathematics. The author has also sent along images of their
work.
Article Title: Maximal monotonicity via convex
analysis
Authors:
Borwein, JM
Journal: J CONVEX ANAL
Volume: 13
Issue: 3-4
Page: 561-586
Year: 2006
* Dalhousie Univ, Fac Comp Sci, Halifax, NS, Canada.
* Dalhousie Univ, Fac Comp Sci, Halifax, NS, Canada.
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
FA—nowadays called the Fitzpatrick function, which was
largely overlooked until the last decade—associated with a monotone
operator A, and similarly a monotone operator Gf 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