Five recent book reviews
CR geometry is nowadays a very broad subject with its scope spanning from the geometric theory of partial differential equations and microlocal analysis to complex and symplectic geometry and foliation theory. This book by Giuseppe Zampieri does not aim to introduce all topics of current interest in CR geometry. Instead, it attempts to be friendly to the novice by moving in a fairly relaxed way from elements of the theory of holomorphic functions in several complex variables to advanced topics of modern CR geometry. The choice of topics provides a good balance between a first exposure to CR geometry and subjects representing current research.
The first chapter of the book covers classical results in several complex variables (Cauchy formulas in polydiscs, Hartogs’ theorems on separability and extendability of holomorphic functions and the logarithmic supermean of the Taylor radius of holomorphic functions). It finishes with the L2 and subelliptic estimates for the ∂-bar operator. The second chapter covers real/complex structures and real/complex symplectic spaces. The Frobenius and Darboux theorems are proved. The third chapter, which constitutes the second half of the book, covers CR structures. A particular emphasis is devoted to analysis of the conormal bundle to a real submanifold of Cn under a canonical transformation. The author then describes the theory of analytic discs attached to real submanifolds and their infinitesimal deformations. The problem of construction of lifts and partial lifts of analytic discs is also addressed. Zampieri also deals with Bloom-Graham normal forms and separate real analyticity. The book is written in a very readable style; every section starts with a short summary and there are a lot of notes and remarks providing a broader context for discussed questions. The reader is supplied with a lot of exercises (hints are provided) and also with suggested research topics, where the author discusses open problems related to the material of the chapter.
The exposition of the subject of algebraic geometry in this book does not pass through sheaf theory, cohomology, derived functors, categories or abstract commutative algebras but it is rather focused on specific examples, together with a part of the basic formalism that is most useful for computations. In particular, Gröbner bases are introduced quite early and for almost every technique there is both an algorithmic and a computational approach. All core techniques of algebraic geometry are developed systematically from scratch, with necessary commutative algebra integrated to geometry. Classical topics (like resultants and elimination theory) are discussed in parallel with affine varieties, morphisms and rational maps. Important examples of projective varieties (like Grassmannians, Veronese and Segre varieties) are emphasised, along with the matrix and exterior algebras needed to write down their defining equations.
This book provides an updated summary of many of the recent developments in the arithmetic of elliptic curves and introduces the reader to the author’s new striking contributions to the subject.
The present collection of papers contains 14 of 72 papers published separately in three volumes under the title Number theory for the Millennium and presented at the Millennium Conference on Number Theory held at Urbana-Champaign. Thirteen of the papers are devoted to concrete mathematical topics related to the „simple“ or multiple Riemann zeta function (Huxley, Matsumoto), the Riemann hypothesis (Balazard), normal numbers (Harman), arithmetical aspects of the theory of curves (Poonen, Perrin-Riou), Diophantine approximation (Tijdeman), the Pell equation (H.Williams), expansion of a given function into a continued fraction (Lorentzen), Waring’s problem (Vaughan and Wooley), pure and mixed exponential sums (Cochrane and Zheng), authomorphic forms (Winnie Li), or to primes in arithmetical progressions (Hooley). Exceptional in this aspect is the paper ‘G. H. Hardy as I knew him’ by R. A. Rankin. All papers certainly fulfil the editors’ hope that a separate publication can help to stimulate the interest in the presented topics or related areas. All of them give the reader an up-to-date information on the development of basic ideas which paved the road to the described major achievements in the subject so that this collection give an integrated picture on main streams in contemporary number theory. As such, it can be recommended to active number theorists and also to a general mathematical audience.
The book under review is the second of four parts of the second edition of the book, which was successfully published a quarter of a century ago. However, the second edition contains some new parts. The book treats a lot of games with their winning strategies, and it is full of pictures and diagrams for reader’s comfort. Although the topic belongs to recreational mathematics, all problems are studied in a very precise way. I am not quite sure, whether it was a good idea to divide the book into four parts. Volume II contains many references to Volume I. Hence it is difficult to read it, if the reader is not familiar with the first volume.