[Editorial note: During the next few weeks, each of the editors of the Mathematical Investor will provide, in an essay format, some personal background explaining the origins of their interest and work in this area. This is a perspective essay by Jonathan M Borwein.]
Early interest in economics and finance
I went to Oxford in 1971 to study functional analysis and number theory but ended up (very quickly) working in Optimization theory with Michael Dempster, who even then was running models of the full non-defence US budget. My early history is described in a chapter of a forthcoming book. My two theses are also on line:
- Optimization with respect to partial orderings (DPhil, Oxon, 1974)
- Monotone operators and nonlinear functional analysis (MSc, Oxon, 1972)
In the 1970’s the modern theory of optimization was still being fleshed out and many of the now familiar ideas were being digested. Many of the most salient names were those of Mathematical economists: Arrow, Aumann, Blackwell, Gale, Nash, Sen to mention just a few — many of the future winners of the Nobel memorial prize in economics.
The next 35 years
I was deeply impressed by especially the work of Sen (qualitative and sensible) and Gale, from whom I learned the approach to convex and nonlinear optimization, which permeates my three texts written decades later but born in that period:
- 2010 on Convex Functions: Constructions, Characterizations and Counterexamples (Cambridge University Press)
- 2005 on Techniques of Variational Analysis (Canadian Mathematical Society/Springer-Verlag)
- 2000/2005 on Convex Analysis and Nonlinear Optimization (Canadian Mathematical Society/Springer-Verlag)
In 1979 I accepted a position at Carnegie Mellon and for a few years interacted daily with economists and finance specialists, operations researchers and public policy experts. I had the good fortune to participate in Alan Newell and Herb Simon’s seminar. By then I had taught linear and non-linear programming before and after the advent of modern personal computing.
I observed then and now, that the strongest Optimization PhDs usually went to mathematics departments, the next tranche went to business schools, and the third cohort to the private sector. I also observed that the salaries went up as the quality of research went down.
I have since seen a lot of the academic world, both as a member of many Faculties, and as a visitor and scientific administrator. (See my CV.) I returned to Canada in 1982, first for another decade at Dalhousie, then briefly in Combinatorics and Optimization in Waterloo, then a decade as Shrum Chair of Science at Simon Fraser University and four years in Computer Science at Dalhousie University. I also held two Canada Research Chairs Between 2001 and 2009.
Finally, in 2009 I assumed a position as Laureate Professor and Director of the Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) at the university of Newcastle in NSW Australia. Throughout all this time I have pursued research in optimization, analysis, number theory and in experimental mathematics, which I feel has given me an optimal view of pure and applied realms, with healthy doses of computing in each.
Recent related research work
My Newcastle research centre’s web pages include ones on visualization and experimentation in mathematics and on my current focus on projection and reflection methods for non-convex inverse problems. I am also an advocate of the Applications of convex analysis within pure mathematics and of Experimental applied mathematics.
Above all I see mathematics in duality: wonderful and pure in its own right but tightly coupled with an external objective reality. (CAVEAT: I know all the terms are loaded and highly charged.) My general views on where mathematics is going are to be found in The Future of Mathematics (1965-2065) which is to appear in a Mathematical Association of America centenary volume in 2015.
Mathematicians and finance
Since before I returned to Canada in 1982, I had developed a healthy scepticism about the role of highly technical quantitative mathematics—both in the hands of decision makers who could not understand the nature of the tools they were deploying, or in the hands of theorists who never actually used the tools in situ.
Nothing I have learned in the next thirty years has changed my mind. It is as import to teach our students the limits of quantification and modelling as it is to show them the power of abstraction when married to intelligent computation.
I remember having to teach business mathematics out of books with interest tables stopping at five percent while inflation was in the twenties. Books which did not even try to explain what they were doing. I have taught future primary school mathematics teachers from books with 47 heuristic principles and sub-principles to learn (a la Polya).
I remember giving courses on linear programming in which full chapters were dedicated to pivot methods which saved a few bits — as was hugely important on the machines of 35 years ago but had little intellectual content or long-term value.
Those ventures were as foolish as they were typical. In my own books, I have tried to avoid such epistemological blunders!
I believe that mathematicians like all other scientists have to monitor the uses and abuses of their ideas. It is not enough to say that all ideas can be used both for good and bad. If, as with much personal financial advice, much bad is being done and is being sanctified with our implicit blessing, we are obliged to speak out. That is the basis of my participation in this group.
Sins of omission are usually less obvious than sins of commission. But they are often even more damaging for being less obvious.