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Book Reviews ReviewsMartin AignerMarkov’s Theorem and 100 Yearsof the Uniqueness ConjectureA mathematical journey fromirrational numbers to perfectmatchingsSpringer International Publishers,Cham, 2013, x + 257 pp.ISBN print: 978-3-319-00887-5ISBN e-book: 978-3-319-00888-2Reviewer: Franz Lemmermeyer (Jagstzell, Germany)The Newsletter thanks Zentralblatt MATH and Franz Lemmermeyerfor the permission to republish this review, originallyappeared as Zbl 1276.00006.Let α be a real number. Dirichlet showed that there exist infinitelymany fractions p q ∈ Q with |α− p q |≤ 1 . Now considerq 2all positive real numbers L such that |α − p q | < 1/Lq2 holdsfor infinitely many fractions p q. The supremum of all these Lis denoted by L(α); then L(α) = 0 if and only if α is rational,and L(α) ≥ 1 otherwise. The set L = {L(α) :α ∈ RQ}is called the Lagrange spectrum. A. Markoff [Math. Ann. 15,381–407 (1879); 17, 379–400 (1880)] showed that the Lagrange√spectrum below 3 consists of all numbers of the form9m2 − 4/m, where m runs through the “Markoff numbers”.These are defined as the set of all natural numbers x i occurringas solutions of the Diophantine equation x 2 1 + x2 2 + x2 3 =3x 1 x 2 x 3 . It is easy to show that every Markoff number appearsas the largest number in some Markoff triple (x 1 x 2 x 3 ), and theuniqueness conjecture predicts that each Markoff number isthe maximum of a unique Markoff triple.The investigation of the uniqueness conjecture from differentperspectives is the main goal of this book. After someintroductory chapters, the reader is introduced to Cohn matrices,the modular group SL(2, Z), free groups, graphs andtrees, and to partial results towards the uniqueness conjectureusing the arithmetic of quadratic fields. The whole discussionis very elementary, and requires no preliminaries except somefamiliarity with basic concepts of algebra and number theory.The reviewer regrets that the author has given in to thetemptation of keeping the book on a very elementary levelthroughout. Readers enjoying the section on hyperbolic geometrywill be well advised to have a look at the verynice book “Fuchsian groups” by S. Katok [Fuchsian groups.Chicago: The University of Chicago Press (1992)]. Similarly,Markoff’s original motivation for studying these questions,the theory of binary quadratic forms, is only briefly mentionedon pp. 36–38 even though quadratic forms cast theirshadows almost everywhere in this book: continued fractionsand the uni-modular group are intimately connected with Lagrangereduction of quadratic forms, and the arithmetic ofideals in quadratic number fields also is just one way of presentingGauss composition and the class group of forms. Thereaders will find a beautiful introduction to the dictionary betweenthese languages in the recent book “Algebraic theoryof quadratic numbers” by M. Trifkovič [Algebraic theory ofquadratic numbers. New York, NY: Springer (2013)].This beautiful book gives readers a chance to familiarizethemselves with a very simple and yet very difficult problemin number theory, and teaches them that it pays to look at aproblem from many different angles. I recommend it to all studentswho are already hooked to number theory, and perhapseven more to those who are not.Franz Lemmermeyer received his Ph.D. fromthe University of Heidelberg and is currentlyteaching mathematics at the gymnasium St.Gertrudis in Ellwangen. His interests includenumber theory and the history of mathematics.New journal from theNew in2014L’Enseignement MathématiqueOrgane officiel de la Commission internationale de l’enseignement mathématiqueISSN print 0013-8584 / ISSN online 2309-46722014. Vol 60, 2 issues. Approx. 450 pages. 17 x 24 cmPrice of subscription: 198 Euro (online only) / 238 Euro (print+online)European Mathematical Society Publishing HouseSeminar for Applied Mathematics, ETH-Zentrum SEW A27CH-8092 Zürich, Switzerlandsubscriptions@ems-ph.org / www.ems-ph.orgAims and Scope:The Journal was founded in 1899 by Henri Fehr (Geneva) and Charles-Ange Laisant (Paris). It is intended primarily for publicationof high-quality research and expository papers in mathematics. Approximately 60 pages each year will be devotedto book reviews.Editors:Anton Alekseev, David Cimasoni, Daniel Coray, Pierre de la Harpe, Anders Karlsson, Tatiana Smirnova-Nagnibeda, András Szenes (all Université deGenève, Switzerland), Nicolas Monod (EPFL, Switzerland), John Steinig, and Vaughan F. R. Jones (University of California at Berkeley, USA)58 EMS Newsletter June 2014