Representation theory (RT) has deep roots in algebra and geometry, and connections with other branches of mathematics, such as algebraic combinatorics and mathematical physics, etc. RT has many subdisciplines, and, more often than not, today we have infinite-dimensional representations of finite-dimensional objects coming to the forefront, as well as the study of representations of infinite-dimensional objects. In this academic year, we are inviting active researchers in RT who often work on the interactions of different fields as indicated above. They will give introductory lectures to bring graduate students, postdocs to new exciting directions, and/or present their own views on recent interesting developments.

Over the past twenty years the subject of low-dimensional topology has been deeply influenced by the ideas from representation theory and from mathematical physics. The purpose of the program on low-dimensional topology at the IMS is to invite leading researchers in these areas to collaborate and to give series of lectures presenting their current research. In particular, the areas of interest are topology in dimensions 3 and 4, gauge theory, topological quantum field theory. The program will encourage the participation of graduate students and young researchers.

Beginning in the 1950s with portfolio theory, modern mathematics has become increasingly involved in the world of finance. A new level of sophistication and understanding came in 1973 with the introduction of the Black-Scholes-Merton partial differential equations and far-reaching probabilistic principles in pricing options. Today it is possible that the formation of yet another level, especially involving modern probability, is underway. The IMS emphasis year on Probability and Mathematical Finance was highlighted by two conferences, the first, "Financial Derivatives, Asset Pricing and Probability Theory," and the second conference, "The Variance Gamma and Related Financial Models."