Five recent book reviews
This is the second volume of the proceedings of the Automorphic Semester held at Centre Emile Borel in Paris in 2000. It contains five important articles on various arithmetic aspects of automorphic forms on GSp(4): G. Laumon extends his earlier results on cohomology of Siegel 3-folds to the case of non-trivial coefficient systems; R. Weissauer constructs ℓ-adic Galois representations attached to discrete series automorphic forms GSp(4) and establishes some of their properties; E. Urban discusses the local behaviour of these representations at ℓ; A. Genestier and J. Tilouine prove a modularity lifting theorem for 4-dimensional symplectic Galois representations; D. Whitehouse proves a weighted twisted fundamental lemma necessary for establishing a functorial transfer of packets from GSp(4) to GL(4). The book will be of interest to researchers and graduate students interested in arithmetic applications of automorphic forms.
The style of this book is very friendly to the reader and the book is obviously well equipped to serve its main purpose, i.e. the exposition of the main kinds of combinatorial designs to undergraduates. As one can expect, the topics include balanced incomplete block designs, their development by means of difference sets, latin squares, one-factorization, tournaments, Steiner triple systems and their large sets, Hadamard matrices and Room squares. There is a section on the Bruck-Ryser-Chowla theorem (in fact, there are two different proofs). Interactions with other parts of mathematics are limited. Of course, one needs to develop the basic notions of affine and projective geometries (there is a section on ovals) and one needs to be able to work with matrices and quadratic residues. Besides matrices, all other necessary notions are explained in the book. That is usually done at the first point where the notion is needed, which makes it possible for the student to get quickly to the objects he or she is interested in. The book is equipped with exercises, there are quite a few historical remarks and almost no claims are stated without proof. Information about unsolved problems is limited, which perhaps makes the book a little less exciting than it could be.
Analysis in metric spaces has considerably expanded in recent years. Experts in analysis on (weighted) Euclidean spaces, manifolds, Carnot groups, and fractals, who are interested in function spaces, harmonic analysis, geometry of curves or quasiconformal geometry, have observed that a ground for some of their methods can be reduced to metric structures and thus the theory can be developed once for all frameworks.
The book by Ambrosio and Tilli presents a representative part of the fundamentals of this development. Part of the text gives elements of measure and integration theory: Riesz representation theorem, weak convergence of measures, construction of measures and particularly Hausdorff measures, abstract integration (this for general increasing set functions by the method of De Giorgi and Choquet). Next, Lipschitz mappings are studied and results related to area and coarea formula are discussed. The part on the geodesic problem presents Busemann's existence result. This claims the existence of a geodesic (i.e., shortest) connection between two points x,y in a metric space E provided all bounded sets of E are compact and there exists a connection of finite length. It is also shown that it is equivalent to consider the connections as connected sets (measured by the one-dimensional Hausdorff measure) or as parameterized curves. This requires some nontrivial facts on rectifiability. The Gromov-Hausdorff convergence of metric spaces is introduced and compactness and embedding results are proved. The method of Gromov-Hausdorff convergence is used to prove an existence theorem for the Steiner problem, which is a generalization of the geodesic problem. Even in the geodesic case, the assumptions are further weakened. The theory of Sobolev spaces of Hajlasz type on metric spaces is developed. It is shown that this concept generalizes the ordinary Sobolev spaces on extension domains. Inequalities of Sobolev and Poincaré type are proved.
The book is based on lectures given by the authors at the Scuola Normale in Pisa and presents a well-chosen selection of what to say to students in the field, when time is limited, with suggestions for further studies. The enjoyable exposition of the subject is supplemented by exercises of various levels.
The famous French mathematician Laurent Schwartz (born on March 15, 1915) died on July 14, 2002. The book contains articles written by Schwartz's colleagues, former students, relatives and friends. Some articles are devoted to mathematical questions and can be appreciated mainly by mathematicians; some others describe Schwartz's numerous political and social activities, his private interests and fancies, concerns and events of his private life. Laurent Schwartz became famous as the author of the theory of distributions, a branch of mathematics that has made its way to all branches of mathematical analysis and applied mathematics. He worked at the University of Paris from 1953, and moved to the École Polytechnique in 1959. His students recall him as an excellent teacher. He had a passion for teaching (he taught his younger brother in childhood to decline Greek nouns with enjoyments on both sides). Schwartz was also involved in a reform of the system of education, including the system for selection of students entering universities. He was also very active in social and political questions. He always supported human rights and actions against wars and colonialism. When the first Mathematical Congress after the war took place in Harvard in 1950, Schwartz had serious troubles getting an entry visa to the USA. After a big effort from mathematicians, a visa was given to him (as well as to J. Hadamard, which was Schwartz's condition).
The mathematician Laurent Schwartz had always made a big effort to improve the world he lived in. He was the chairman of a protest association “Comité Audin”, investigating the murder of a mathematician Maurice Audin during the Algerian war in the 50’s, and a founding member of the “Comitée des mathématiciens” in the 70’s, which was created to support all persecuted mathematicians in totalitarian countries. He was also engaged in many other social activities. Shortly before his death, he published his autobiography “Un mathématicien aux prises avec le siècle”, which is an excellent description and analysis of development of mathematics and of the political events in the past century.
Nonlinear spectral theory is a relatively new field of mathematics, which is far from being complete, and many fundamental questions still remain open. The main focus of this book is therefore formulated by the authors as the following question: How should we define a spectrum for nonlinear operators in such a way that it preserves useful properties of the linear case but admits applications to a possibly large variety of nonlinear problems? Contrary to the linear case, the spectrum of a nonlinear operator contains practically no information on this operator. The authors convince the reader that it is not the intrinsic structure of the spectrum itself, which leads to interesting applications, but its property of being a useful tool for solving nonlinear equations. The book is an excellent presentation of the “state-of-the-art” of contemporary nonlinear spectral theory as well as a glimpse of the diversity of directions in which current research is moving.
The whole text consists of 12 chapters. The authors recall basic facts on the spectrum of a bounded linear operator in the first chapter. In Chapter 2, some numerical characteristics providing quantitative descriptions of certain mapping properties of nonlinear operators are studied. The classical Kuratowski measure of noncompactness plays a key role here. Chapter 3 is devoted to general invertibility results. In particular, conditions that guarantee that the local invertibility of a nonlinear operator implies its global invertibility are of interest. The Rhodins and the Neuberger spectra are studied in Chapter 4. In Chapter 5, the authors study a spectrum for Lipschitz continuous operators, first proposed by Kachurovskij in 1969, and a spectrum for linearly bounded operators, introduced recently by Dörfner. Chapter 6 discusses the spectrum for certain special continuous operators introduced by Furi, Martelli and Vignoli in 1978, and its modification introduced recently by Appell, Giorgieri and Väth. The Feng spectrum is discussed in detail in Chapter 7. Chapter 8 is devoted to the study of “local spectrum” due to Väth, which in the literature is called “phantom”. In Chapter 9, the authors investigate a modification of the Feng spectrum proposed by Feng and Webb and another spectrum introduced by Singhof-Weyer and Weberand Infante-Webb. Chapter 10 is devoted to the study of nonlinear eigenvalue problems. The authors concentrate on the notion of a “nonlinear eigenvalue”, nonlinear analogue of the Krein-Rutman theorem, connected eigenvalues, etc. Chapter 11 contains a description on how numerical ranges may be used to localize the spectrum of a nonlinear operator on the real line or in the complex plane. Selected applications are presented in the last Chapter.
The exposition of nonlinear spectral theory in this book is self-contained. All major statements are proved; each definition and notion is carefully illustrated by examples. To understand this text does not require any special knowledge and only modest background of nonlinear analysis and operator theory is required. The book is addressed not only to mathematicians working in analysis but also to non-specialists wanting to understand the development of spectral theory for nonlinear operators in the last 30 years. The bibliography is rather exhaustive and so this text will certainly serve as an excellent reference book for many years. I am convinced that at least one copy of this book should be in any mathematical library.