Five recent book reviews
General module theory is too vast an area to provide for general structure theory, the major problem being the lack of a satisfactory decomposition theory. In this monograph, the authors propose the notions of a natural class of modules (i.e. a class closed under submodules, direct sums and essential extensions) and of a type submodule (i.e. a submodule maximal among those belonging to a natural class) to overcome this problem. They call a module M a TS-module if every type submodule of M is a direct summand. TS-modules thus generalize the notions of an extending module or a CS-module. In chapter 4, type dimension theory of a module is developed in analogy to the classical finite uniform dimension theory. Chapter 5 extends decomposition theory of CS-modules to TS-modules. It contains moreover a far reaching generalization of the Goodearl-Boyle decomposition theory of non-singular injective modules into type I, II, and III submodules (using the fact that in the appropriate generalization, type I, II, and III modules form a natural class each). Chapter 6 deals with relations between the structure of a ring R and the lattice of (pre-)natural classes of R-modules. As the authors point out, the book not only presents a new theory but it suggests a new path through general ring and module theory.
This book covers in an encyclopedic way many aspects of the broad concept of ‘distance’. It is divided into seven parts and 28 chapters, the majority of the text dealing with a rich variety of metrics on an equally rich class of mathematical objects. Some space is devoted to entries from outside of mathematics, including concepts as foreign to quantification as for example ‘moral distance’ or curiosities like ‘Ironman distance’ or Hollywood ‘co-starring distance’ (the last one to be found in chapter 22, which is called “Distances in Internet and Similar Networks”). The title correctly announces that the book is written as a dictionary, that is, the reader is given a list of entries, each of them treated in a short and succinct way. The collection originates from a personal archive of the authors. The majority of the book is written for a reader who is familiar with technical mathematical language. At the end of the book is a list of entries but there is no index of keywords (that means for example that you will not find the Euler angle metric if looking for the keyword ‘angle’; it is listed only under the letter ‘E’). Anybody who wants a comprehensive and reliable list of different mathematical (and some non-mathematical) concepts of distance in one place (and without using Google) will find it in this book.
This book is the second volume of a two-volume handbook reviewing many topics of contemporary differential geometry. In this volume there are eight surveys on various themes: Finsler geometry and its differential invariants (written by J. Alvarez Paiva), foliations, characteristic classes and deformation theory (by R. Barre and A. Kacimi Alaoui), symplectic manifolds, Lagrangian submanifolds and complex structures, Hamiltonian geometry and symplectic reduction (by A. Cannas da Silva), metric Riemannian geometry including Gromov-Hausdorff distance, collapsing Riemannian manifolds, Alexandrov spaces and Hausdorff convergence (by K. Fukaya), contact manifolds, particularly in dimension 3 (by H. Geiges), complex manifolds, Kahler manifolds and harmonic differential forms (by I. Mihai), Lagrange spaces, Finsler spaces and Lagrange spaces of higher order (by R. Miron), and Lorentzian manifolds and their curvature, geodesics and the Bochner techniques (by F. Palomo and A. Romero). The volume therefore covers many important fields in differential geometry.
This rather short book gives an introduction to all important aspects of the theory of elliptic curves. As a consequence, many numerically or algebraically demanding calculations are left to the reader, often in the form of exercises. The book starts with a discussion of some analytic aspects of elliptic curves (including elliptic integrals, elliptic functions and projective realizations of elliptic curves). A more algebraic approach is used in a description of the standard correspondence between equivalence classes of elliptic curves and lattices, thereby leading to the j-function. By the end, the author has turned to more advanced topics like counting points on elliptic curves, curves with complex multiplication and the use of modular forms for proving the Jacobi formula for the number of representations of a positive integer as a sum of four squares. The book will be useful both for students of mathematics and computer science.
This small booklet contains lecture notes on the Calogero-Moser integrable system and its relation to other fields of mathematics. In one small place, the reader will find a fascinating kaleidoscope of interesting and modern topics, all interacting with the main theme of the book. The author offers a short description of each subject involved, subjects that include Poisson geometry and Hamiltonian reduction; integrable systems, action-angle variables and the classical and trigonometric Calogero-Moser system; deformation theory of associative algebras and Hochschild cohomology, together with Kontsevich quantization of Poisson structures; quantum momentum maps, quantum Hamiltonian reduction and the Levasseur-Stafford theorem; quantization of the Calogero-Moser system and the Calogero-Moser systems for finite Coxeter groups; Dunkl operators and Olshanetsky-Perelomov Hamiltonians; the rational Cherednik algebra and its relation to double affine Hecke algebras; symplectic reflection algebras and the Koszul deformation principle; the deformation approach to symplectic reflection algebras and, finally, representation theory for rational Cherednik algebras. The exposition still keeps the flavour of the corresponding lectures and it presents in a nice, condensed but understandable form the core of the theory. It really is a remarkable book.