Five recent book reviews
The main topic of this book is combinatorial and differential topology. The author discusses a lot of interesting and basic facts avoiding sophisticated techniques, hence the reading of the book requires only a modest background for these topics (e.g. basic topological properties of sets in Euclidean space). After an introductory discussion of graphs, the topology of subsets in Euclidean space is considered (including the Jordan theorem for curves, the Brouwer fixed point theorem and the Sperner lemma). Simplicial complexes and CW-complexes are discussed in the next chapter, followed by a treatment of surfaces, coverings, fibrations and homotopy groups. The fifth chapter turns to differential topology (smooth manifolds, embeddings and immersions, the degree of a map, the Hopf theorem on the homotopy classification of maps to the sphere and Morse theory). The last chapter treats the fundamental groups (with many explicit examples). The book contains a lot of problems and their solutions can be found at the end of the book.
Emil Artin was one the leading personalities of algebra and number theory from the early part of the 20th century. He was the only mathematician to solve two of the famous Hilbert problems and many important notions now bear his name: artinian rings, the Artin reciprocity law, Artin L-functions, etc. The present volume of 'History of Mathematics - Sources' reprints a selection of Artin's work: his three famous short books (The Gamma Function, Galois Theory and Theory of Algebraic Numbers) and ten papers, mainly on algebra (on braid groups, real fields etc.), three of which appear here in English for the first time. Michael Rosen has written an introduction providing interesting comments on the reprinted work and a brief biographical sketch starting from Artin's childhood in Reichenberg (Liberec) and covering his emigration to the USA and his final return to Hamburg.
Random surfaces, or random height functions, are random functions defined on Z d or Rd and taking values (heights) in Z or R. Their distributions are determined by Gibbs potentials invariant under a lattice of translations and depending only on height differences. This is a general framework that covers many particular models considered in the literature. In this setting, a variational principle is proved, namely invariant Gibbs measures of given slope are those of minimal specific free energy. Continuous models are approximated by discrete ones with increasing resolution and a large deviation principle is proved. New results concerning the uniqueness of the Gibbs state are presented. The book concludes with a list of open problems.
This is a book of mathematical pictures. It contains about 1000 illustrations of curves and surfaces of standard mathematical functions created by the computer algebra system Mathematica. In the introductory chapter the basic concepts concerning curves and surfaces are given. It also includes a modern classification of curves and surfaces. The structure of the following sixteen chapters is such that the formulas and/or the specific values of the parameters of the curves are on the left page while the corresponding plots are on the right page. New or reorganized sections in this edition are Green’s functions, minimal surfaces, knots, links, Archimedean solids, duals of Platonic solids and stellated forms. With the exception of the introduction all the chapters are on the accompanying CD-ROM in the form of Mathematica notebooks. In fact, there is more material on the CD than in the book. Moreover, some omissions in the printed version are corrected in the electronic one. The reader can use the Mathematica notebooks to modify and play with plots of all the functions presented in the book. This book is recommended to anybody interested in the field.
This voluminous book is intended as an elementary introduction aimed at undergraduate university students. Each section states (without proof) relevant theorems and then proceeds by means of simple examples and exercises. An important feature is that the exposition is richly accompanied by computer algebra code (equally distributed between Matlab, Mathematica and Maple). The major part of the book is devoted to classical linear theory (both for systems and higher order equations). The necessary material from linear algebra is also covered. More advanced topics include numerical methods (Euler, Runge-Kutta), stability of equilibria, bifurcations, Laplace transforms and the power series method. The elementary character of the book makes it accessible to a wide audience of students; it also serves as a simple introduction to the above mentioned computer programs.