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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

Scientific prize winners of the ICM 2010

The scientific prizes of the International Mathematical Union of the year 2010 were awarded during the Opening Ceremony of ICM 2010 in Hyderabad, India on August 19, 2010 to the following scientists ... more ...


“How far off the edge of the table can we reach by stacking n identical, homogeneous, frictionless blocks of length 1?” is a famous elementary, but nontrivial question pertaining to sequences.
László Leindler recalls in his review of Mike Paterson's and Uri Zwick's
article “Overhang” [Am. Math. Mon. 116, No. 1, 19-44 (2009; Zbl 1168.40001)] that a “classical solution achieves an overhang asymptotic to 1/2 lnn. This solution is widely believed to be optimal.” more ...

Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

“The book under review, preserving the physical motivations, has been written to develop, step by step, the techniques and methods used when groups (emphasizing the representations) are applied to describe physical phenomena, with details and explanations that are usually omitted in most textbooks,“ begins Rutwig Campoamor-Stursberg's review of the book
“Lie groups, physics and geometry. An introduction for physicists, engineers and chemists” [Cambridge: Cambridge University Press (2008; Zbl 1157.00009)] by Robert Gilmore. more ...

Elementary number theory. Primes, congruences, and secrets.

“Oh no, not yet another introduction to elementary number theory!” - Franz Lemmermeyer's review of William Stein's “Elementary number theory. Primes, congruences, and secrets. A computational approach” [Undergraduate Texts in Mathematics. New York, NY: Springer (2009; Zbl 1155.11002)] starts with irony, but comes quickly to its advantages: more ...

Modeling differential equations in biology

Applications to biology make up an essential part of mathematical models used in today's praxis. Clifford Henry Taubes' “Modeling differential equations in biology” [2nd ed. Cambridge: Cambridge University Press. (2008; Zbl 1156.34002)] is a

“...delightfully written textbook [which] was prepared using the materials developed by the author for a course on applications of differential equations in biological sciences at Harvard University and is aimed at introducing biologists to a wide variety of ideas and techniques that are useful in mathematical modeling of biological systems. The prerequisites are limited to the fundamentals of calculus (good skills in differentiation and integration would be very helpful), and additional material is introduced in the text whenever necessary.” more ...

Quantum information theory and quantum statistics

It has been a long way since C. E. Shannon started to explore the mathematical background of communcation and information theory [Bell Syst. Tech. J. 27, 379-423, 623-656 (1948; Zbl 1154.94303)]. Today, the vast development in understanding quantum phenomena offers impressive new perspectives and applications, many of them surveyed Dénes Petz's “Quantum information theory and quantum statistics” [Theoretical and Mathematical Physics. Berlin: Springer (2008; Zbl 1145.81002)]. more ...

Logarithmic forms and Diophantine geometry

“The quantity of recent results quoted in this book reveals the intense vitality of the subject,”

summarizes Michel Waldschmidt his extensive review of Alan Baker's and Gisbert Wüstholz's “Logarithmic forms and Diophantine geometry” [New Mathematical Monographs 9. Cambridge: Cambridge University Press. (2007; Zbl 1145.11004)]. more ...

The Painlevé handbook

“The new monograph of the leading experts in the singularity analysis of differential equations provides a profound introduction to the Painlevé property and related topics on the boundary between integrable and nonintegrable differential and difference models. Explaining their point of view, the authors follow the principal line: first specify equations which pass or partially pass the Painlevé test for conditions which are assumed to be necessary for integrability, and then, using a singular part transformation and its extensions, integrate the obtained equation, or prove its integrability, or at least construct particular solutions to the equation,“

begins Andrei A. Kapaev's review of The Painlevé handbook [Dordrecht: Springer. (2008; Zbl 1153.34002)] by Robert M. Conte and Micheline Musette. more ...

Einstein's mistakes

“Albert Einstein was certainly the greatest physicist of the 20th century, and one of the greatest scientists of all time. It may be presumptuous to talk of mistakes made by such a towering figure, especially in the centenary of his annus mirabilis. But the mistakes made by leading scientists often provide better insight into the spirit and presuppositions of their times than do their successes.”

With this quote from Steven Weinberg's article Einstein's mistakes begins the review of Svitlana P. Rogovchenko of Hans C. Ohanians, “Einstein’s mistakes. The human failings of genius.” [New York, NY: W. W. Norton & Co. (2008; Zbl 1156.01012)], which we like to single out here as a somewhat twisted congratulation to his 130th birthday. more ...

Zeta functions of groups and rings

For a long time, zeta functions have been playing an exceptional role in propelling mathematical research and connecting seemingly different objects.

“The purpose of this stimulating book is to bring into print significant and as yet unpublished work from different areas of the theory of zeta functions of groups. There are numerous calculations of zeta functions of groups which are yet to be made into print. These explicit calculations provide evidence in favour of conjectures, or indeed can form inspiration and evidence for new conjectures”, summarizes Andrea Lucchini his review of Marcus du Sautoy's and Edward Woodward's “Zeta functions of groups and rings” [Lecture Notes in Mathematics 1925 (2008; Zbl 1151.11005)], and recalls: more ...

Hedgefunds. An analytic perspective.

“...risk has become ‘endogenous’ in certain markets - particularly those that have recently become flush with large inflows of assets - which is one of the reasons that the largest players can no longer assume that historical estimates of volatility and price impact are accurate measures of risk exposure”, concludes Andrew W. Lo in August 2007 as one result of his research program, carried out for almost ten years and condensed in his book “Hedgefunds. An analytic perspective.” [Princeton, NJ: Princeton University Press (2008; Zbl 1151.91003 )]. more ...

The Bogomolov conjecture for totally degenerate abelian varieties

In two subsequent publications [Invent. Math. 169, No. 2, 321-376 (2007; Zbl 1153.14036); Invent. Math. 169, No. 2, 377-400 (2007; Zbl 1153.14029)], Walter Gubler proves the Bogomolov conjecture in the case where the abelian variety is totally degenerate at some place of the ground field. more ...

The Mathematics of voting

The intricacies of a faithful representation of the popular vote are subject of a detailed investigation in three recent publications more ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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.

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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.

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.
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Abel prize 2010
I. M. Gelfand 1913-2009

Scientific prize winners of the ICM 2010
Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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