My research interests have shifted dramatically in the last few years. I would like to give some background to that change.
I have always been interested in the applications of mathematics. In fact, my earliest career aspirations evolved from becoming an engineer, to becoming a physicist, and then to becoming a mathematician. My first inclinations there were applied as well. I worked for two summers while in high school in the actuarial department of the London Life Insurance Company in London, Ontario.
I entered the University of Waterloo with an actuarial career in mind, and early in my undergraduate studies took the first actuarial exam. But Waterloo has very broad mathematical offerings, and I also had early exposure to the abstract side of things. I found that I liked that even more, and by the end of my undergraduate days I was becoming familiar with topics such as universal algebra, group representation theory, algebraic geometry, and category theory. Along the way, I maintained my interest in the financial applications of mathematics by taking several courses in economics.
At the University of Chicago, I knew I was interested in working in algebra or a related field, but it took me a while to narrow it down. I started in the direction of algebraic topology, then explored commutative algebra in some depth (with one resulting publication), and ended by writing a dissertation in the relatively new field of Hopf algebras.
As a faculty member, I continued to work primarily in Hopf algebras, but pursued other mathematical interests as well, with commutative algebra being the secondary field in which I published most frequently. Financial economics became a side interest. I read Friedman and Modigliani, struggled with Keynes, and subscribed to the publications of the Federal Reserve Bank of St. Louis. I read the seminal Black-Scholes paper; I could not follow the stochastic calculus at that point, but I programmed the formula for use with my personal investments.
I am quite proud of the achievements I have made in pure mathematics. However, I decided several years ago that I would like to take up the financial applications direction that I had considered in the past, and thus turn my side interest into my professional interest as well. The opportunity to do so was available because the financial mathematics program at FSU was by then well underway.
This has required a massive amount of re-education. I needed to augment my interdisciplinary knowledge, and so sat in on graduate courses in Statistics and Economics, and participated in a doctoral-level seminar in Finance. During summers, I spent all of my time learning or re-learning the mathematics used in finance, reading books about the applications, and trying to work out for myself how it all went together. My most rapid progress was made during my sabbatical year at Carnegie Mellon University, where I was able to participate in courses in their Master of Science in Computational Finance program such as Multiperiod Asset Pricing, Numerical Methods for Finance, Stochastic Calculus for Finance, Simulation Methods for Option Pricing, and Term Structure of Interest Rates, and also participate in the advanced graduate seminar and the year-long course in Advanced Stochastic Calculus and interact with the faculty concerning research questions.
At this point I have a good background in financial mathematics, although, of course, the more one knows, the more one realizes how much there is that one does not know. As I have in the past, I find that there are quite a number of areas that I would like to explore more deeply, perhaps with colleagues or students. One general topic is the interplay between discrete and continuous models. Do the continuous models give a "complex approximation" to a discrete world, or do they represent "beautifully simple truth"? Or, perhaps, does it not matter? I began at CMU an examination of a particular discrete approximation of the beautiful Heath-Jarrow-Morton interest rate model; intriguing possibilities remain to be explored, but the difficulties seem substantial. Another project involving options led to a dead end (my idea was simply wrong), but I remain quite interested in exploring the area. Although much of financial mathematics focuses on the very short term, I have been interested since my London Life days in questions with a long time horizon.
Starting in Summer 2010, I have begun looking at the "radically elementary" approach to probability theory initiated by Edward Nelson. This uses finite probability spaces, but in a different way than is usually done, and seems to have the same scientific content as conventional theories that involve far more elaborate mathematical structures. As I have always thought that finance was essentially discrete anyway, I am very excited in seeing if these methods lead to a simpler and better understanding of financial issues.
Besides my financial interests, I still sometimes do other work on the side. The paper "Algebraic Myhill-Nerode Theorems", co-written with Robert Underwood, has appeared in the journal Theoretical Computer Science, and there is certainly room for a sequel.