Banach space theory is a basic tool of mathematical analysis that has been of importance since its inception at the turn of the century. In recent years, there has been a rebirth of interaction with other aspects of mathematical analysis -- such as harmonic analysis and operator theory -- and probability theory. In particular, Banach space methods, in the hands of highly creative and powerful researchers, have led to fundamental breakthroughs in these other fields. World centers of excellence in this area are in Israel, Paris, Warsaw, Ohio, and Texas. Professor Phelps has had a long and distinguished career in Banach space theory, his emphasis being on convexity analysis and differentiability. His recent work, however, has emphasized applications outside of Banach space theory -- to statistical mechanics, optimization, and approximation. His theoretical work led him to problems in lattice gases, important to statistical mechanics, in which he proved differentiabilty of the statistical mechanics pressure function, and to interesting work on conjugate projections and nearest point maps in the theory of optimization. He has also obtained (negative) results for researchers in mathematical economics. All of these applications are made possible by his expertise in the "pure" theory, and they bring evidence to the importance of basic mathematical research for applied problems. In the current proposal, Professor Phelps plans to investigate a number of related problems concerning the geometry of convex subsets of Banach spaces. In particular, he will consider the validity in complex spaces of the classical Bishop-Phelps theorem on the density of support functionals, as well as other problems related to Asplund spaces and the Mazur intersection property. Research is also to be conducted by two postdoctoral mathematicians, Professors Rychlik and Zhu. Professor Zhu is a new Ph.D. whose work in operator theory addresses a variety of aspects of several complex variables. He will continue his research on Toeplitz and Hankel operators on Bergman spaces of bounded symmetric domains and the relation to functions of bounded mean oscillation. Professor Rychlik, an expert in ergodic theory and dynamical systems, will focus his research on the development of Poincare-type expansions near a saddle of a vector field in n-dimensions. He will also continue his work on the dynamics of the Lorentz and Henon equations.
