A Life in Mathematics
When I was 19 years old, in spite of being a poor student, I decided that I was going to earn a PhD in mathematics. The audacity of this decision is a surprise to me even now as I look back at this long career. How does someone with such mediocre mathematical talents manage to survive a grueling graduate program, become a professor in a research-intensive department and contribute to the mathematical establishment?
There are two reasons that come to mind. The first is the training that a student of mathematics receives. The insistence on precision and logical thinking is excellent preparation to address, not only mathematics, but also problems that society encounters. The second reason is more personal, more a reflection of my personality. I work very hard, I am determined, I am curious, I am blessed with a small amount of creativity, and most important of all, I am enthusiastic about whatever I do. I am fortunate to have received this training. It transformed my life.
One lesson that I learned is that mathematical knowledge can be minimal in order to carry out mathematical research. My undergraduate preparation was just terrible, yet when I arrived in graduate school, after two years in Vietnam, I blossomed. This same lesson is applied daily in mathematics departments in the US as undergraduates work on research problems. Perhaps the challenge of addressing problems on your own is more important than course work in developing mathematical talent.
I enjoy problems. Problems sharpen the intuition (Consider: Let f be a polynomial with integer coefficients. Then f can be considered as a function on Q or R. Question: If f is 1-1 as a function on Q, is it 1-1 as a function on R?). Publishable research is the currency of our profession. How do we find problems that are interesting, solvable and worthy of publication?
[Photo: Velez in 1965]