Search

Noncommutative differential geometry is a novel approach to geometry, aimed in part at applications in physics. It was founded in the early eighties by the 1982 Fields Medalist Alain Connes on the basis of his fundamental works in operator algebras. It is now a very active branch of mathematics with actual and potential applications to a variety of domains in physics ranging from solid state to quantization of gravity. The strategy is to formulate usual differential geometry in a somewhat unusual manner, using in particular operator algebras and related concepts, so as to be able to plug in noncommutativity in a natural way. Algebraic tools such as K-theory and cyclic cohomology and homology play an important role in this field. It is an important topic both for mathematics and physics.

This volume is a collection of articles by speakers at the conference “Poisson 2006: Poisson Geometry in Mathematics and Physics”, which was held June 5–9, 2006, in Tokyo, Japan. Poisson 2006 was the fifth in a series of international conferences on Poisson geometry that are held once every two years. The aim of these conferences is to bring together mathematicians and mathematical physicists who work in diverse areas but have common interests in Poisson geometry. The program for Poisson 2006 was remarkable for the overlap of topics that included deformation quantization, generalized complex structures, differentiable stacks, normal forms, and group-valued moment maps and reduction. The articles represent current research in Poisson geometry and should be valuable to anyone interested in Poisson geometry, symplectic geometry, and mathematical physics. This volume also contains lectures by the principal speakers of the three-day school held at Keio University that preceded Poisson 2006.

Noncommutative geometry is a novel approach which is opening up new possibilities for geometry from a mathematical viewpoint. It is also providing new tools for the investigation of quantum space-time in physics. Recent developments in string theory have supported the idea of quantum spaces, and have strongly stimulated the research in this field. This self-contained volume contains survey lectures and research articles which address these issues and related topics. The book is accessible to both researchers and graduate students beginning to study this subject.

Hideki Omori is widely recognized as one of the world's most creative and original mathematicians. This volume is dedicated to Hideki Omori on the occasion of his retirement from Tokyo University of Science. His retirement was also celebrated in April 2004 with an in?uential conference at the Morito Hall of Tokyo University of Science. Hideki Omori was born in Nishionmiya, Hyogo prefecture, in 1938 and was an undergraduate and graduate student at Tokyo University, where he was awarded his Ph.D degree in 1966 on the study of transformation groups on manifolds [3], which became one of his major research interests. He started his?rst research position at Tokyo Metropolitan University. In 1980, he moved to Okayama University, and then became a professor of Tokyo University of Science in 1982, where he continues to work today. Hideki Omori was invited to many of the top international research institutions, including the Institute for Advanced Studies at Princeton in 1967, the Mathematics Institute at the University of Warwick in 1970, and Bonn University in 1972. Omori received the Geometry Prize of the Mathematical Society of Japan in 1996 for his outstanding contributions to the theory of in?nite-dimensional Lie groups.

In honour of the 65th birthday of Professor M Obata a workshop was held at Keio University. This volume includes notes on the talks and discussions which took place and cover a wide range of subjects on geometry, global analysis, topology and mathematical physics.

This volume contains the refereed proceedings of two symposia on symplectic geometry and quantization problems which were held in Japan in July 1993. The purpose of the symposia was to discuss recent progress in a range of related topics in symplectic geometry and mathematical physics, including symplectic groupoids, geometric quantization, noncommutative differential geometry, equivariant cohomology, deformation quantization, topological quantum field theory, and knot invariants. The book provides insight into how these different topics relate to one another and offers intriguing new problems. Providing a look at the frontier of research in symplectic geometry and quantization, this book is suitable as a source book for a seminar in symplectic geometry.