I like good mathematics that solves nontrivial problems. The intellectual pursuit of this ideal led me through a sometimes stochastic journey from China to this country.
My undergraduate major in China was electrical engineering, and in my sophomore year, I skipped the rest of grades and went into graduate studies in pure mathematics because I was so attracted to highest abstraction. My MS thesis was on radicals and socles of modules, in which I proved a Szasz-type structure theorem on semi-primitive rings that satisfy the decreasing chain condition on principal ideals. Frustrated by unknown utility of my theorem, I teamed up after my Master degree with three gifted, young algebraists to study computational complexity theory. We would get together several times a week, in marathon seminars, lasting many hours. We were not afraid of attacking the problem of P versus NP, and we tried it continuously in multi-year, but fruitless efforts. Mathematics in that group of infatuated young people was very ambitious.
After coming to U.S. in 1989, I was pressured by friends who cared about my employment prospects, and decided to do something different, and something more practical. I began to take courses at random, but got the most satisfaction from taking probability classes and from my PhD dissertation studies on stochastic convexity and stochastic majorization. Probability is a fundamental way of viewing the world that is driven by both order and chance. Central to statistical data analysis, probability becomes commonplace in physics, genetics, geosciences, neuroscience, engineering, etc. The application of probability to finance has revolutionized the entire global financial industry.
Probability is also a core mathematical discipline, and in addition to algebraic, ordering and topological structures, probability provides another fundamental structure that enriches mathematical thought processes. Many of the most complex deterministic phenomena can actually be best analyzed through the probabilistic lens. The probabilistic method is a central approach in the analysis of algorithms and computational complexity. Probabilistic representations give new insights into a wide variety of PDEs from existence of solutions to their regularity properties. Most intriguingly, some very disordered sets, such as the set of prime numbers, can be decomposed into a highly structured part and a part that behaves in a highly random fashion.
Probability also provides me with academic and consulting opportunities. I have been on both the mathematics faculty and statistics faculty, teaching both mathematics and statistics courses and advising both mathematics and statistics students. My current research focuses on high-dimensional risk analysis, which is at the intersection of probability, statistics, game theory and optimization. This is an emerging area where one can find good mathematics that solves nontrivial problems.
For details on my research, visit my faculty webpage at http://www.math.wsu.edu/math/faculty/lih/