Mathematical Sciences: Geometry of Convex Sets
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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.
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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