Shigeyuki Morita

Toshikazu Sunada

Kenji Ueno

Kōji Shiga

Three volumes originating from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students.

The Interplay Between Topology, Functions, Geometry, and Algebra

A Mathematical Gift, III

Pavel Exner

Jonathan P. Keating

Peter Kuchment

Alexander Teplyaev

This book addresses a new interdisciplinary area emerging on the border between various areas of mathematics, physics, chemistry, nanotechnology, and computer science. The focus here is on problems and techniques related to graphs, quantum graphs, and fractals that parallel those from differential equations, differential geometry, or geometric analysis. Also included are such diverse topics as number theory, geometric group theory, waveguide theory, quantum chaos, quantum wire systems, carbon nano-structures, metal-insulator transition, computer vision, and communication networks. This volume contains a unique collection of expert reviews on the main directions in analysis on graphs (e.g., on discrete geometric analysis, zeta-functions on graphs, recently emerging connections between the geometric group theory and fractals, quantum graphs, quantum chaos on graphs, modeling waveguide systems and modeling quantum graph systems with waveguides, control theory on graphs), as well as research articles.

Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007

Analysis on Graphs and Its Applications

This is the first of three volumes originated from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students. The main purpose of the lectures was to show the listeners the beauty and liveliness of mathematics using the material that is accessible to people with little preliminary knowledge. The first chapter of the book talks about the geometry and topology of surfaces. Among other topics the authors discuss the Poincare-Hopf theorem about critical points of vector fields on surfaces and the Gauss-Bonnet theorem about the relation between the curvature and topology (Euler characteristics). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincare approach to dimension. It also discusses the structure of three-dimensional manifolds, proving, in particular, that the three-dimensional sphere is the union of two doughnuts.

A Mathematical Gift, I

Geometry in ancient Greece is said to have originated in the curiosity of mathematicians about the shapes of crystals, with that curiosity culminating in the classification of regular convex polyhedra addressed in the final volume of Euclid’s Elements. Since then, geometry has taken its own path and the study of crystals has not been a central theme in mathematics, with the exception of Kepler’s work on snowflakes. Only in the nineteenth century did mathematics begin to play a role in crystallography as group theory came to be applied to the morphology of crystals. This monograph follows the Greek tradition in seeking beautiful shapes such as regular convex polyhedra. The primary aim is to convey to the reader how algebraic topology is effectively used to explore the rich world of crystal structures. Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal. For that reason the title Topological Crystallography has been chosen. Topological crystals can be described as “living in the logical world, not in space,” leading to the question of how to place or realize them “canonically” in space. Proposed here is the notion of standard realizations of topological crystals in space, including as typical examples the crystal structures of diamond and lonsdaleite. A mathematical view of the standard realizations is also provided by relating them to asymptotic behaviors of random walks and harmonic maps. Furthermore, it can be seen that a discrete analogue of algebraic geometry is linked to the standard realizations. Applications of the discussions in this volume include not only a systematic enumeration of crystal structures, an area of considerable scientific interest for many years, but also the architectural design of lightweight rigid structures. The reader therefore can see the agreement of theory and practice.

With a View Towards Discrete Geometric Analysis

Topological Crystallography

Motoko Kotani

This volume is an outgrowth of an international conference in honor of Toshikazu Sunada on the occasion of his sixtieth birthday. The conference took place at Nagoya University, Japan, in 2007. Sunada's research covers a wide spectrum of spectral analysis, including interactions among geometry, number theory, dynamical systems, probability theory and mathematical physics. Readers will find papers on trace formulae, isospectral problems, zeta functions, quantum ergodicity, random waves, discrete geometric analysis, value distribution, and semiclassical analysis. This volume also contains an article that presents an overview of Sunada's work in mathematics up to the age of sixty.

International Conference on the Occasion of Toshikazu Sunada's 60th Birthday, August 6-10, 2007, Nagoya University, Nagoya, Japan

Spectral Analysis in Geometry and Number Theory

The Taniguchi Symposium on global analysis on manifolds focused mainly on the relationships between some geometric structures of manifolds and analysis, especially spectral analysis on noncompact manifolds. Included in the present volume are expanded versions of most of the invited lectures. In these original research articles, the reader will find up-to date accounts of the subject.

Proceedings of the 21st International Taniguchi Symposium held at Katata, Japan, Aug. 23-29 and the Conference held at Kyoto, Aug. 31 - Sep. 2, 1987

Geometry and Analysis on Manifolds

Masaki Maruyama

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.

Toshikazu Sunada

About

Books

Join the club