In 1929, Mr. Toyosaburo Taniguchi established the Taniguchi Foundation with the goal of promoting research in the basic sciences in Japan and to engender mutual understanding on an international level via the exchange of ideas and research. In 1956, he instituted a division for mathematics within the Foundation and sponsored the first summer seminar. Since that time, the seminar has been held each year on various mathematical topics. In 1974, Mr. Taniguchi promoted and sponsored an International Symposium in various fields of science on a smaller scale. His aim was to raise the level of scientific thought and research, while providing a forum where promising young scholars the world over could gather informally to exchange thoughts and to contribute their knowledge.These gatherings were held until 1999. This volume is a collection of the research manuscripts written by the invited speakers at the final conference set up by the Taniguchi Foundation, the Taniguchi Conference on Mathematics 1998, held in Nara, Japan. The conference was aimed at gathering all previous participants of Taniguchi Symposia.
The subject areas were chosen to include all important and active fields of mathematics. Hence the topics in this volume are quite diverse. The contributors are world-class mathematicians who are generally reporting on subjects for which they are well known. For example, contributions include R. E. Borcherds on vertex algebras, M. Kontsevich on non-commutative algebraic manifolds, P.-L. Lions on fluid mechnics, M. Kashiwara on micro-localization, J. Kollar on the topology of algebraic varieties, S. Mori on rational curves in algebraic varieties, and others.
Hardy spaces, Carleson measures and a gradient estimate for harmonic functions on negatively curved manifolds by H. Arai Quantum vertex algebras by R. E. Borcherds Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds-Summary by K. Fukaya and K. Ono Very singular diffusion equations by M.-H. Giga, Y. Giga, and R. Kobayashi The topology of real and complex algebraic varieties by J. Kollar Approximation of expectation of diffusion process and mathematical finance by S. Kusuoka Representation theory in characteristic $p$ by G. Lusztig Probability and geometry by P. Malliavin The intrinsic Hodge theory of $p$-adic hyperbolic curves by S. Mochizuki Operator algebras, topology and subgroups of quantum symmetry--Construction of subgroups of quantum groups by A. Ocneanu Large deviation and hydrodynamic scaling by S. R. S. Varadhan.