Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Geometry Ramanujan‘s Lost Notebook An Introduction to Mathematical of the American Mathematical Society Selected Topics in Plane and Solid Part II Cryptography May 2008 Volume 55, Number 5 Geometry G. E. Andrews, Penn State University, University J. Hoffstein, J. Pipher, J. Silverman, Brown J. Aarts, Delft University of Technology, Park, PA, USA; B. C. Berndt, University of Illinois University, Providence, RI, USA Mediamatics, The Netherlands at Urbana, IL, USA This self-contained introduction to modern This is a book on Euclidean geometry that covers The “lost notebook” contains considerable cryptography emphasizes the mathematics the standard material in a completely new way, material on mock theta functions—undoubtedly behind the theory of public key cryptosystems while also introducing a number of new topics emanating from the last year of Ramanujan’s life. and digital signature schemes. The book focuses Interview with Martin Davis that would be suitable as a junior-senior level It should be emphasized that the material on on these key topics while developing the undergraduate textbook. The author does not mock theta functions is perhaps Ramanujan’s mathematical tools needed for the construction page 560 begin in the traditional manner with abstract deepest work more than half of the material in and security analysis of diverse cryptosystems. geometric axioms. Instead, he assumes the real the book is on q- series, including mock theta Only basic linear algebra is required of the numbers, and begins his treatment by functions; the remaining part deals with theta reader; techniques from algebra, number theory, introducing such modern concepts as a metric function identities, modular equations, and probability are introduced and developed as space, vector space notation, and groups, and incomplete elliptic integrals of the first kind and required. The book covers a variety of topics that thus lays a rigorous basis for geometry while at other integrals of theta functions, Eisenstein are considered central to mathematical the same time giving the student tools that will series, particular values of theta functions, the cryptography and is an ideal introduction for be useful in other courses. Rogers- Ramanujan continued fraction, other q- mathematics and computer science students to continued fractions, other integrals, and parts of the mathematical foundations of modern 2008. Approx. 410 p. 300 illus. (Universitext) Hecke’s theory of modular forms. cryptography. The book includes an extensive Softcover bibliography and index; supplementary materials ISBN 978-0-387-78240-9 approx. $49.95 2008. Approx. 430 p. 3 illus. Hardcover are available online. ISBN 978-0-387-77765-8 approx. $89.00 Surveys in Number Theory 2008. Approx. 540 p. 65 illus. (Undergraduate Texts in Mathematics) Hardcover K. Alladi, University of Florida, Gainesville, FL, Naive Lie Theory ISBN 978-0-387-77993-5 $49.95 USA (Ed.) J. Stillwell, University of San Francisco, CA, USA Number theory has a wealth of long-standing This book aims to fill a gap in the literature by Multiscale Methods problems, the study of which over the years has introducing Lie theory to junior and senior level led to major developments in many areas of undergraduates. In order to achieve this, the Averaging and Homogenization mathematics. This volume consists of seven author focuses on the so-called “classical groups,’’ G. Pavliotis, Imperial College,London, London, significant chapters on number theory and viewed as matrix groups with real, complex, or England; A. Stuart, University of Warwick, related topics. Written by distinguished quaternion entries. This allows them to be London, England mathematicians, key topics focus on multiparti- studied by elementary methods from calculus Multiscale Methods begins by setting the tions, congruences and identities (G. Andrews), and linear algebra. Each chapter is enhanced theoretical foundations of the methods and then the formulas of Koshliakov and Guinand in with numerous exercises, discussion of further moves on to develop models and prove Ramanujan’s Lost Notebook (B. C. Berndt, Y. Lee, results, and historical comments. theorems. Numerous examples show how to and J. Sohn), alternating sign matrices and the apply multiscale methods to solving a variety of 2008. Approx. 225 p. 25 illus. (Undergraduate Texts Weyl character formulas (D. M. Bressoud), theta problems. Exercises allow the reader to build in Mathematics) Hardcover functions in complex analysis (H. M. Farkas), their own skills and put them into practice. representation functions in additive number ISBN 978-0-387-78214-0 approx. $49.95 theory (M. B. Nathanson), and mock theta 2008. Approx. 330 p. (Texts in Applied Mathematics, Volume 55, Number 5, Pages 553–656, May 2008 functions, ranks, and Maass forms (K. Ono), and Volume 53) Hardcover elliptic functions (M. Waldschmidt). ISBN 978-0-387-73828-4 $49.95 2008. Approx. 200 p. (Developments in Mathematics, Volume 17) Hardcover ISBN 978-0-387-78509-7 approx. $69.95 Sign up for New Book Alerts at springer.com Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Distribution Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice. All prices are net prices. 013649x Patterns in factorization (see page 595) Trim: 8.25" x 10.75" 112 pages on 40 lb Cougar Opaque • Spine: 1/4" • Print Cover on 9pt Carolina ,!4%8 ,!4%8 ,!4%8 Pro A MERIC A N M ATHEM ATIC A L S OCIETY STUDENT MATHEMATICAL LIBRARY SERIES Books in the AMS series, Student Mathematical Library (STML), will spark students’ interests in modern mathematics and increase their appreciation for research. This series features well-written, challenging, expository works that capture the fascination and usefulness of Higher Arithmetic mathematics. STML series books are An Algorithmic Introduction to Number Theory suitable for course adoption and/or Harold M. Edwards, New York University, NY self-study. 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Volume 37; 2007; 152 pages; Softcover; ISBN: 978-0-8218-4220-1; List US$29; AMS members US$23; Order code STML/37 Finite Fields and Applications Gary L. Mullen, Pennsylvania State University, University Park, PA, and Carl Mummert, University of Michigan, Ann Arbor, MI An accessible introduction to the theory of finite fields that focuses heavily on fascinating applications View the latest STML series catalog at This book is co-published with Mathematics Advanced Study Semesters. www.ams.org/bookstore/newpubscatalogs Volume 41; 2007; 175 pages; Softcover; ISBN: 978-0-8218-4418-2; List US$35; AMS members US$28; Order code STML/41 1-800-321-4AMS (4267), in the U. S. and Canada, or 1-401-455-4000 (worldwide); fax:1-401-455-4046; email: [email protected]. American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA New & Forthcoming from Birkhäuser Algebra, Arithmetic, and Frames and Bases An Introduction to the Geometry An Introductory Course Heisenberg Group and the In Honor of Y.I. Manin OLE CHRISTENSEN, Technical University of Denmark, Lyngby, Denmark Sub-Riemannian Isoperimetric YURI TSCHINKEL, New York University, NY, USA; YURI G. ZARHIN, Pennsylvania State University, University Park, Based on a streamlined presentation of the author’s Problem PA, USA (Eds) previous work, An Introduction to Frames and Riesz LUCA CAPOGNA, University of Arkansas, Fayetteville, AR, USA; DONATELLA DANIELLI, Purdue University, West The two volumes of Algebra, Arithmetic, and Geometry: Bases, this new textbook fi lls a gap in the literature, Lafayette, IN, USA; SCOTT D. PAULS, Dartmouth College, In Honor of Y.I. Manin consist of invited expository developing frame theory as part of a dialogue between Hanover, NH, USA; JEREMY T. TYSON, University of articles and surveys detailing Manin’s contributions to mathematicians and engineers. Newly added sections Illinois, Urbana-Champaign, IL, USA the subjects. on applications will help mathematically oriented readers to see where frames are used in practice and This book provides an introduction to the basics of In celebration of Manin’s 70th birthday, the following engineers to discover the mathematical background for sub-Riemannian differential geometry and geometric well-respected and distinguished mathematicians have applications in their fi eld. The primary focus of the work analysis in the Heisenberg group, focusing primarily contributed: K. Behrend, V. Berkovich, J. Bost, P. Bressler, is on basic topics without too many side remarks making on the current state of knowledge regarding Pierre D. Calaque, J. Carlson, A. Chambert-Loir, E. Colombo, it easier for students to follow the presentation. Pansu’s celebrated 1982 conjecture regarding the A.