Highlights in Zentralblatt MATH

In this new column you have quick access to selected highlights of Zentralblatt MATH.
You will find summaries on remarkable developments in mathematics, outstanding reviews, historical surveys and much more, as compiled by the editors of Zentralblatt MATH for your advantage.

Date	Highlights
2008-12-29	Introduction to modern cryptography. “The book is a wide and complete survey of modern cryptography. Its emphasis is on the information-theoretical analysis of the methods, but several applications are introduced in a well motivated manner. The exercises are illustrative of the methods and pose important challenges not only to novel readers but also to crypto professionals profiting from the book as a reference source. The book is highly recommended as a textbook in cryptography courses at graduate or advanced undergraduate levels”, begins Guillermo Morales-Luna his review of “ Introduction to modern cryptography.” by Jonathan Katz and Yehuda Lindell [Chapman & Hall/CRC Cryptography and Network Security. Boca Raton, FL (2008; Zbl 1143.94001)]. more ...
2008-12-02	Moonshine beyond the monster The Monstrous Moonshine conjecture, proved by R. E. Borcherds [Invent. Math. 109, 405--444 (1992; Zbl 0799.17014)], became famous at least since Borcherds' Fields medal in 1998. Terry Gannon's book “Moonshine beyond the monster. The bridge connecting algebra, modular forms and physics” [Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. (2006; Zbl 1146.11026)] gives, in Hiromichi Yamada's words, “an overview of the current status of Moonshine and a perspective on future study in mathematics and physics.” more ...
2008-11-11	A random approach to the Lebesgue integral Most people meet integrals for the first time in connection with Riemann sums, just to discover later the power of more general approaches. Naturally, the question of a possible convergence of random Riemann sums for Lebesgue integrable functions arises. Jack Grahl [J. Math. Anal. Appl. 340, No. 1, 358-365 (2008; Zbl 1147.28001)] proves that the random Riemann sums of a sequence of partitions whose sizes tend to 0 converge in probability to ∫f. As Olav Nygaard mentions in his review, more ...
2008-10-31	Advanced algebra. Along with a companion volume ‘Basic algebra’ “All together, this is another outstanding textbook written by the renowned and versatile mathematical researcher, teacher, and author Anthony W. Knapp that reflects his spirit, his devotion to mathematics, and his rich experiences in expository writing at best”, summarizes Werner Kleinert his review of Anthony W. Knapp's “Advanced algebra.” (Zbl 1133.00001) more ...
2008-10-10	Beyond partial differential equations “Although the text was written for truly advanced graduate students, it contains a wealth of well presented results on semigroup theory and the theory of evolution equations with applications to hyperbolic equations and systems and will therefore serve as a valuable resource for researchers in mathematics and theoretical physics as well,” reads Thomas Hagen's review of the book “Beyond partial differential equations. On linear and quasi-linear abstract hyperbolic evolution equations.” by Horst Reinhard Beyer [Lecture Notes in Mathematics 1898. Berlin: Springer. (2007; Zbl 1144.35001)]. more ...
2008-09-20	100 Years of Hardy's “A Course of Pure Mathematics” “Fifty one years ago I bought my copy of this book and it has been in use ever since....My book is the 1944 ninth edition, a copy of the 1938 seventh edition that was the first revision and resetting of the original 1908 work.” Peter S. Bullen looks back to many decades of mathematics in his Zentralblatt review of the Centenary edition of Hardy's book. It happens rarely that a mathematical textbook remains almost unchanged in use for such a period, which clearly indicates the foresight of Hardy's concepts. more ...
2008-08-31	Bourbaki reprint Some days ago, Henri Cartan passed away, the last living founder of the famous Bourbaki group. In the course of the last years, the full extent of the “Éléments de mathématique” was made available again as a reprint of the English translation. Most volumes of this edition are already reviewed in Zentralblatt, with a special emphasis of the second reviews on the historical significance and the development of the reception during the decades.
2008-08-15	D-modules, perverse sheaves, and representation theory. D-modules bridge many different areas of mathematics, including Algebraic and Differential Geometry, Complex Analysis, Category Theory, Differential Equations, and Representation Theory. “D-modules, perverse sheaves, and representation theory” is an expanded translation of a Japanese monograph of Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki. As the review of Gheorge Gussi [Zbl 1136.14009] points out: more ...
2008-07-13	History of Banach spaces and linear operators Looking backward to the unfolding of a new mathematical branch often provides deep insights into its concepts, methods and interconnections. This is especially true for the area of Banach spaces and linear operators, topic of a new remarkable book of Albrecht Pietsch. more ...
2008-06-12	L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld In the context of the local Langlands correspondence, a geometric relation between the Lubin-Tate tower and the Drinfeld tower has long been expected. The recent book “L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld” of Laurent Fargues, Alain Genestier and Vincent Lafforgue makes this relation explicit. Read the elaborated review of Elmar Große-Klönne in Zbl 1136.14001 to get acquainted with the sophisticated mathematics involved. more ...
2008-05-30	PISA, Bach, Pythagoras: Mathematics, Music and Cabaret We learned from Tom Lehrer that mathematics and cabaret can be a good match. Jürgen Appell reviews in Zbl 1134.00008 the program of Dietrich Paul, a German cabaret artist whose favorite topics evolve around education, music and mathematics. more ...
2008-05-13	Electronic Geometry Models ZMATH includes now the items of the Electronic Geometry Models archive, like the Rhombicosidodecahedron (one of the thirteen Archimedean solids, with 60 vertices, 120 edges and 62 facets, 12 pentagons, 30 squares and 20 triangles), or Sharir's cube. more ...
2008-05-09	Factorization method in quantum mechanics Special problems from quantum mechanics that arise frequently in physics and chemistry can often be treated by the factorization method. more ...
2008-04-17	Does a global temperature exist? Global warming has become a hot topic in the mass media. »Warming« is usually measured by the change of temperature - but does a »global temperature« make sense? more ...
2008-03-31	The proof of the Poincaré and the Geometrization Conjecture Read the first reviews of Grisha Perelman's three famous e-prints, provided by Gérard Besson (Grenoble) in Zbl 1130.53001, Zbl 1130.53002 and Zbl 1130.53003.
2008-03-29	Who is Alexander Grothendieck? On the occasion of Grothendieck's 80th birthday, ZMATH has an extensive review by Werner Kleinert (Berlin) of the first part of Winfried Scharlau's new biography in Zbl 1129.01018.