This project will support research on probability problems that arise from a wide variety of fields. Topics from genetics include regulatory sequence evolution, gene duplication, and the analysis of data generated by cancer genome projects. In finance, understanding bubbles in markets, swing options, and credit risk associated with subprime mortgages require sophisticated mathematical ideas from control theory, backward stochastic differential equations, and enlargement of filtration. The study of financial markets, insurance risk, and communication and computer networks lead to stochastic models that exhibit long range dependence and heavy tails. The analysis of these processes and the new phenomena they present, involves techniques much different from the classical theory of independent random variables, making use of ideas from a wide variety of sources including ergodic theory. In addition, there is a fertile interaction between probability and a variety of topics that arise from algebra, geometry, and analysis, including the use of analytic techniques to study convergence rates of Markov chains.
Probability theory, born in the analysis of gambling games, now plays an important role in biology, physics, economics, finance, insurance, communication networks, and in many topics within mathematics. The main aim of this proposal is to train more researchers in probability to tackle the many important problems that arise from its applications. Cornell is an ideal place for these activities because of the strong traditions of research in probability and interdisciplinary work in applied mathematics. Support for summer research projects will show undergraduates that probability is an interesting research area with many important applications. At the graduate level, increased fellowship support will enhance the training of our Ph.D. students. Postdoctoral positions will help new graduates expand their horizons, while hot topics conferences and annual summer schools will be a national resource for broadening the education of researchers (young and old). In all of these activities, the focus will be on developing theory to treat problems that arise from applications.
