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Zbl 1188.00001
Colmez, Pierre
Elements of analysis and algebra (and number theory). (Éléments d'analyse et d'algèbre (et de théorie des nombres).) (French)
[B] Palaiseau: Les Éditions de l'École Polytechnique. 469~p. EUR~35.00 (2009). ISBN 978-2-7302-1563-3/pbk

The present book grew out of a course designed for upper-undergraduate students of mathematics at École Polytechnique, Paris, France. Building upon a solid background knowledge of basic undergraduate mathematics, it offers an introduction to three more advanced fields of fundamental importance, thereby preparing the reader for further in-depth studies in various areas of contemporary mathematics. More precisely, the three mathematical theories discussed in this volume concern the representations of finite groups and their applications, the basics of functional analysis, and the elements of one-dimensional complex analysis. The core material is enhanced by six appendices devoted to related topics, both classical and modern, which not only serve as a beautiful illustration of the unity of mathematics as a whole, but also provide some fascinating insight into recent developments and problems in modern mathematics.\par The text starts with a mathematical vocabulary. This part is basically a preparatory chapter recalling (and regrouping) the fundamentals of group theory, linear algebra, general topology, real functions, normed vector spaces, and $p$-adic numbers.\par This summary is utmost detailed and well-arranged, with full proofs of the main results being given, and it contains a wealth of related exercises, for many of which complete solutions are provided at the end of this introductory part.\par Chapter I deals with representations of finite groups and their characters. The material is developed as a natural extension of basic linear algebra, on the one hand, and as a first approach to the Fourier transform on the other. This chapter ends with the theorems of Artin-Brauer.\par Chapter II provides the first steps into functional analysis by introducing Banach spaces, Hilbert spaces, Fourier series, and -- as a special topic -- Banach spaces over the field $\Bbb Q_p$ of $p$-adic numbers.\par Chapter III turns to integration theory, with the focus on the Lebesgue integral and the related function spaces.\par Chapter IV discusses the theory of the (analytic) Fourier transform in various function spaces, including the theorem of Riemann-Lebesgue and the Poisson formula.\par Chapter V lays the foundations of complex analysis by explaining holomorphic functions of one complex variable, whereas Chapter VI complements the discussion by deriving Cauchy's formula and the calculus of residues.\par Chapter VII, the last chapter of the book, is devoted to the more specific topic of Dirichlet series and its role in number theory. The material covered here comprises the basic properties of Dirichlet series defined in the complex half-plane, the Mellin transform, the integral formula for Dirichlet series, the Riemann zeta function, Dirichlet characters and Dirichlet $L$-functions, Dirichlet's theorem of primes in arithmetic progressions, the Möbius function, and the Ramanujan tau function.\par Appendix A gives a proof of the famous prime number theorem, together with an outlook to the Riemann Hypothesis and to the Lindelöf Hypothesis.\par Appendix B deals with the Haar measure on the topological group $\text{SL}_n(\Bbb R)$ and with the volume of the quotient space $\text{SL}_n(\Bbb R)/\text{SL}_n(\Bbb Z)$.\par Appendix C provides further examples of representations of finite groups, including permutation groups, $p$-groups, and the group $\text{GL}_2(\Bbb F)$ for a finite field $\Bbb F$.\par Appendix D touches upon functions of a $p$-adic variable, with a special emphasis on locally analytic functions on $\Bbb Z_p$ and on the $p$-adic zeta function.\par Appendix E points to the arithmetic of elliptic curves and to the conjecture of Birch and Swinnerton-Dyer, with briefly explaining modular forms along the way.\par Appendix F is very advanced and topical in that it provides an introduction to the celebrated Langlands program in algebraic number theory.\par In the course of this very instructive appendix, which directly leads to the forefront of current research in the field, the reader gets acquainted with adèles and idèles, with the according variants of integral transforms (à la Fourier and Mellin) of $L$-functions, and with automorphic representations.\par Appendix G contains nine selected problems, each of which is divided into several separate parts and questions. These problems are to illustrate how the basic theories developed in the main text can be applied to answer related, further-leading, and very fundamental questions in the respective context. For each of these problems, a detailed solution is worked out. Also, the complexity of these selected problems (and their solutions) is to demonstrate, once more, the unity of mathematics in its entirety.\par The whole text is enriched by a vast amount of complementing, carefully selected and highly instructive exercises. Together with the just as numerous footnotes and accompanying remarks, which put the core material of the book in a much wider context, these exercises essentially extent the fascinating panorama of some of the most central and ubiquitous areas of modern pure mathematics. Overall, both the cultural and the historical aspects of the subjects discussed in the current book are strongly emphasized, and the effective combination of classical topics and recent developments is just as masterful. In this regard, it is fair to state that the introductory textbook under review is of highly original and outstanding character. In fact, it offers a tremendous wealth of fundamental mathematics at its best, with a wide panoramic view, and that in an utmost lucid, detailed, enlightening and inspiring expository style of mathematical writing. No doubt, this book is an extremely useful companion for every student striving for a broad knowledge of fundamental contemporary mathematics. A translation into English of this excellent, fairly unique text would be a great gain for the international mathematical community as a whole.
[Werner Kleinert (Berlin)]

MSC 2000:: *00A05 General mathematics
11-01 Textbooks (number theory)
12-01 Textbooks (field theory)
15-01 Textbooks (linear algebra)
20-01 Textbooks (group theory)
30-01 Textbooks (functions of one complex variable)
11R37 Class field theory for global fields
46-01 Textbooks (functional analysis)
11M38 Zeta and $L$-functions in characteristic $p$
54-01 Textbooks (general topology)
28-01 Textbooks (measure and integration)

Keywords: textbook (general mathematics); group theory; representation theory; functional analysis; complex analysis; Fourier analysis; $p$-adic analysis; Langlands program

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