Heritage of European Mathematics
ISBN 978-3-03719-058-6
October 2009, 661 pages, hardcover, 17 x 24 cm.
Hans Freudenthal (1905?1990) was a Dutch mathematician, born in Luckenwalde, Germany. His scientific activities were of a rich variety. Enrolling at the University of Berlin as a student in the 1920s, he followed in the footsteps of his teachers, and became a topologist, but with a lively interest in group theory. After a long journey through the realm of mathematics, working on almost all subjects that drew his interest, he turned towards the practical and methodological issues of the didactics of mathematics.
The present Selecta are devoted to Freudenthal’s mathematical oeuvre, they contain a selection of his major contributions. Included are fundamental contributions to topology such as the foundation of the theory of ends (in the thesis of 1931), the introduction (in 1937) of the suspension and its use in stability results for homotopy groups of spheres. In group theory there is work on topological groups (of the 1930s) and on various aspects of the theory of Lie groups, such as a paper on automorphisms of 1941. From the later work of the 1950s and 1960s, papers on geometric aspects of Lie theory (geometries associated to exceptional groups, space problems) have been included. Freudenthal’s versatility is further demonstrated by a choice from his foundational and historical work: papers on intuitionistic logic and topology, a paper on axiomatic geometry reappraising Hilbert’s Grundlagen, and a paper summarizing his development of Lincos, a universal (“cosmic”) language.
The book also contains a sketch of Freudenthal’s life. Most of the selected papers are accompanied by brief comments.
EMS Tracts in Mathematics Vol. 9
ISBN 978-3-03719-074-6
DOI 10.4171/074
October 2009, 195 pages, hardcover, 17 x 24 cm.
This book develops a new cohomological theory for schemes in positive characteristic p and it applies this theory to give a purely algebraic proof of a conjecture of Goss on the rationality of certain L-functions arising in the arithmetic of function fields. These L-functions are power series over a certain ring A, associated to any family of Drinfeld A-modules or, more generally, of A-motives on a variety of finite type over the finite field Fp. By analogy to the Weil conjecture, Goss conjectured that these L-functions are in fact rational functions. In 1996 Taguchi and Wan gave a first proof of Goss’s conjecture by analytic methods a la Dwork.
The present text introduces A-crystals, which can be viewed as generalizations of families of A-motives, and studies their cohomology. While A-crystals are defined in terms of coherent sheaves together with a Frobenius map, in many ways they actually behave like constructible etale sheaves. A central result is a Lefschetz trace formula for L-functions of A-crystals, from which the rationality of these L-functions is immediate. Beyond its application to Goss’s L-functions, the theory of A-crystals is closely related to the work of Emerton and Kisin on unit root F-crystals, and it is essential in an Eichler?Shimura type isomorphism for Drinfeld modular forms as constructed by the first author.
The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. It assumes a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories, as provided by standard textbooks. Beyond that the presentation is largely self-contained.
ISBN: 978-0-470-74306-5
Hardcover
200 pages
September 2009
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ISBN: 978-1-4051-9767-0 Paperback
ISBN: 978-1-4051-9766-3 Hardcover
256 pages
November 2009,
Berto’s highly readable and lucid guide introduces students and the interested reader to Godel’s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Godel's arguments.
Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters
Discusses interpretations of the Theorem made by celebrated contemporary thinkers
Sheds light on the wider extra-mathematical and philosophical implications of Godel’s theories
Written in an accessible, non-technical style
400 pages | 79 black and white line illustrations | 234x156mm
978-0-19-533654-2 | Hardback | March 2010 (estimated)
This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, and fractal surfaces. This synthesis will complement currently required courses dealing with these topics and expose the prospective reader to some new and deep relationships. In addition to providing a classical introduction to the main issues involving approximation and interpolation with uni- and multivariate splines, cardinal and exponential splines, and their connection to wavelets and multiscale analysis, which comprises the first half of the book, the second half will describe fractals, fractal functions and fractal surfaces, and their properties. This also includes the new burgeoning theory of superfractals and superfractal functions. The theory of splines is well-established but the relationship to fractal functions is novel. Throughout the book, connections between these two apparently different areas will be exposed and presented. In this way, more options are given to the prospective reader who will encounter complex approximation and interpolation problems in real-world modeling. Numerous examples, figures, and exercises accompany the material.
Readership: Students and scholars of mathematics, partiularly the study of communalities between splines and fractals in interpolation and approximation theory, fractal functions and fractal surfaces
1: The General Interpolation and Approximation Problem
2: Splines
3: Interpolation in Rs, s > 1
4: Fractals
5: Fractal Functions
6: Fractal Surfaces
7: Superfractals
8: Superfractal Functions