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
Letter to the EditorLetter to the EditorMassimo Ferri (Università di Bologna, Italy)The Role of ApplicationorientedMathematicsDear Editor,In the March Newsletter, Ciro Ciliberto, President of theItalian Mathematical Union, complained about the negativecomments on a Marie Curie fellowship proposal inmathematics, concerning a lack of interdisciplinarity andsocio-economic impact of the proposed project. I fullyagree with him on the fact that the evaluation criteriaseem not to comprehend the specificities of pure mathematicalresearch; I hope that it will be possible to givethese criteria more flexibility and also that interdisciplinaritywithin mathematics will be recognised as such.Still, it seems to me that this event stresses a point thathas already appeared in this newsletter in other forms:does mathematics “sell” itself conveniently to the scientificcommunity and society at large? I think it would benecessary to do it not only for getting funds, grants, etc.,but also as a fair attitude of mathematicians toward society.I’m not only speaking of popularisation – which isan important cultural issue anyway – but also of a bettertwo-way communication between mathematics andother scientific and technological areas. There is not just“applied” mathematics but “application-oriented” mathematics,inspired by the many facets of modern technol-ogy, which is growing fast in our departments and canprovide the link I am hoping for.Unfortunately, in my opinion, application-orientedmathematics is not given the correct appreciation withinmathematics when it comes to fund distribution and –above all – academic competition, at least in Italy. Theremay be several reasons for that: the claim that application-inspiredmathematics is of a lower level; the fact thatits correctness is more difficult to check; even a sort ofretaliation against the attitude denounced by ProfessorCiliberto (an attitude which has severe fallout in termsof funding). On the other side, mathematicians often andcorrectly “defend” on the media our discipline, pointingout its manifold applications.I think it’s high time to discuss this issue openly. I seetwo symmetric rigidities: a lack of sensitivity of somescientific environments to the specificities of pure mathematics,and a lack of sensitivity of some mathematicalenvironments to the importance of application-orientedresearch. Should we go on pretending that the problemdoes not exist? If not, how can we face it? Is this just anItalian phenomenon or a European one? I hope that theEMS Newsletter will offer its pages to a debate whichcannot be deferred any longer.Massimo FerriProfessor of Geometry at the University of BolognaNew journal from theNew in2014Annales de l’Institut Henri Poincaré DCombinatorics, Physics and their InteractionsISSN print 2308-5827 / ISSN online 2308-58352014. Vol 1, 4 issues. Approx. 400 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 unfolding of new ideas in physics is often tied to the development of new combinatorial methods, and conversely someproblems in combinatorics have been successfully attacked using methods inspired by statistical physics or quantum fieldtheory. The journal is dedicated to publishing high-quality original research articles and survey articles in which combinatoricsand physics interact in both directions. Combinatorial papers should be motivated by potential applications to physicalphenomena or models, while physics papers should contain some interesting combinatorial development. Both rigorousmathematical proof and heuristic physical reasoning clearly labeled as such have a place in this journal.Editors-in-Chief:Gérard H. E. Duchamp (Univ. Paris XIII, France)Vincent Rivasseau (Univ. Paris XI, France)Alan Sokal (New York Univ., USA and Univ. College London, UK)Managing Editor:Adrian Tanasa (Univ. Paris XIII, France)EMS Newsletter June 2014 59