In the past thirty years, differential geometry has undergone an enormous change with infusion of topology, Lie theory, complex analysis, algebraic geometry and partial differential equations. Professor Matsushima played a leading role in this transformation by bringing new techniques of Lie groups and Lie algebras into the study of real and complex manifolds. This volume is a collection of all the 46 papers written by him.

This book collects the papers published by A. Borel from 1983 to 1999. About half of them are research papers, written on his own or in collaboration, on various topics pertaining mainly to algebraic or Lie groups, homogeneous spaces, arithmetic groups (L2-spectrum, automorphic forms, cohomology and covolumes), L2-cohomology of symmetric or locally symmetric spaces, and to the Oppenheim conjecture. Other publications include surveys and personal recollections (of D. Montgomery, Harish-Chandra, and A. Weil), considerations on mathematics in general and several articles of a historical nature: on the School of Mathematics at the Institute for Advanced Study, on N. Bourbaki and on selected aspects of the works of H. Weyl, C. Chevalley, E. Kolchin, J. Leray, and A. Weil. The book concludes with an essay on H. Poincaré and special relativity. Some comments on, and corrections to, a number of papers have also been added.

In 1996 the AMS awarded Goro Shimura the Steele Prize for Lifetime Achievement :" To Goro Shimura for his important and extensive work on arithmetical geometry and automorphic forms; concepts introduced by him were often seminal, and fertile ground for new developments, as witnessed by the many notations in number theory that carry his name and that have long been familiar to workers in the field." 103 of Shimura ́s most important papers are collected in four volumes. Volume II contains his mathematical papers from 1967 to 1977 and some notes to the articles.

This volume contains selected papers of Dr Morikazu Toda. The papers are arranged in chronological order of publishing dates. Among Dr Toda's many contributions, his works on liquids and nonlinear lattice dynamics should be mentioned. The one-dimensional lattice where nearest neighboring particles interact through an exponential potential is called the Toda lattice which is a miracle and indeed a jewel in theoretical physics. The papers in this volume can be grouped into five subjects: statistical mechanics, theory of liquids and solutions, lattice dynamics, Toda lattice and soliton theory and its applications.

This book provides a detailed description of a most important unsolved mathematical problem — the Goldbach conjecture. Raised in 1742 in a letter from Goldbach to Euler, this conjecture attracted the attention of many mathematical geniuses. Several great achievements were made, but only until the 1920's. The book gives an exposition of these results and their impact on mathematics, particularly, number theory. It also presents (partly or wholly) selections from important literature, so that readers can get a full picture of the conjecture.

This book gives a self-contained fundamental study of the subject. Besides the following special features it contains the author's detailed solution to the long-standing unsolved problem in the theory of complex manifolds: Does there exist a complex structure on the six-sphere? The special features of the book are: a classification of almost complex (and similarly, almost Hermitian) structures together with inclusion relations; discussions about various known almost Hermitian structures; a necessary and sufficient condition for a general almost Hermitian manifold to have constant holomorphic sectional (or bisectional) curvature and similar conditions for various special almost Hermitian manifolds; some complex Laplacians together with some of their relationships with the real Laplacian; the spectral geometry of Riemannian manifolds and some general almost Hermitian manifolds including K„hlerian manifolds as a special case; conditions for an almost complex structure to be a complex structure; some vanishing theorems for Riemannian and almost Hermitian manifolds.

This book provides a detailed description of a most important unsolved mathematical problem OCo the Goldbach conjecture. Raised in 1742 in a letter from Goldbach to Euler, this conjecture attracted the attention of many mathematical geniuses. Several great achievements were made, but only until the 1920''s. The book gives an exposition of these results and their impact on mathematics, particularly, number theory. It also presents (partly or wholly) selections from important literature, so that readers can get a full picture of the conjecture."

This book is devoted to an analysis of the way that structures must enter into a serious study of any subject, and the term “structuralism” refers to the general method of approaching a subject from the viewpoint of structure. A proper appreciation of this approach requires a deeper understanding of the concept of structure than is provided by the simple intuitive notion of structures that everyone posseses to some degree. Therefore, a large part of the discussion is devoted directly or indirectly to a study of the nature of structures themselves. A formal definition of a structure, plus some basic general properties and examples, is given early in the discussion. Also, in order to clarify the general notions and to see how they are used, the later chapters are devoted to an examination of how structures enter into some special fields, including linguistics, mental phenomena, mathematics (and its applications), and biology (especially in the theory of evolution). Because the author is a mathematician, certain mathematical ideas have influenced greatly the choice and approach to the material covered. In general, however, the mathematical influence is not on a technical level and is often only implicit. Even the chapter on mathematical structures is nontechnical and is about rather than on mathematics. Only in the last chapter and earlier in three short sections does one find any of the expected “formal” mathematics. In other words, the great bulk of the material is accessible to someone without a mathematical background.

Translation generalized quadrangles play a key role in the theory of generalized quadrangles, comparable to the role of translation planes in the theory of projective and affine planes. The notion of translation generalized quadrangle is a local analogue of the more global ?Moufang Condition?, a topic of great interest, also due to the classification of all Moufang polygons. Attention is thus paid to recent results in that direction, but also many of the most important results in the general theory of generalized quadrangles that appeared since 1984 are treated.Translation Generalized Quadrangles is essentially self-contained, as the reader is only expected to be familiar with some basic facts on finite generalized quadrangles. Proofs that are either too long or too technical are left out, or just sketched. The three standard works on generalized quadrangles are (co-)authored by the writers of this book: ?Finite Generalized Quadrangles? (1984) by S E Payne and J A Thas, ?Generalized Polygons? (1998) by H Van Maldeghem, and ?Symmetry in Finite Generalized Quadrangles? (2004) by K Thas.

This book deals systematically with the mathematical theory of plane elasto-statics by using complex variable methods, together with many results originated by the author. The problems considered are reduced to integral equations, Fredholem or singular, which are rigorously proved to be uniquely solvable. Particular attention is paid to the subjects of crack problems in the quite general case, especially those of composite media, which are solved by a unified method. The methods used in this book are constructive so that they may be used in practice.