Surveys in Modern Mathematics
Surveys in Modern Mathematics

Author: Victor Prasolov

Publisher: Cambridge University Press

Published: 2005-04-14

Total Pages: 364

ISBN-13: 9781139441124

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This collection of articles from the Independent University of Moscow is derived from the Globus seminars held there. They are given by world authorities, from Russia and elsewhere, in various areas of mathematics and are designed to introduce graduate students to some of the most dynamic areas of mathematical research. The seminars aim to be informal, wide-ranging and forward-looking, getting across the ideas and concepts rather than formal proofs, and this carries over to the articles here. Topics covered range from computational complexity, algebraic geometry, dynamics, through to number theory and quantum groups. The volume as a whole is a fascinating and exciting overview of contemporary mathematics.

Concepts of Modern Mathematics
Concepts of Modern Mathematics

Author: Ian Stewart

Publisher: Courier Corporation

Published: 2012-05-23

Total Pages: 367

ISBN-13: 0486134954

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In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts of groups, sets, subsets, topology, Boolean algebra, and other mathematical subjects. 200 illustrations.

Field Arithmetic
Field Arithmetic

Author: Michael D. Fried

Publisher: Springer Science & Business Media

Published: 2005

Total Pages: 812

ISBN-13: 9783540228110

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Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

Real Algebraic Geometry
Real Algebraic Geometry

Author: Michel Coste

Publisher: Springer

Published: 2006-11-15

Total Pages: 425

ISBN-13: 3540473378

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Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King; C. Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W. Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M. Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P. Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard- Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T. Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M. Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin; T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.

Rational Curves on Algebraic Varieties
Rational Curves on Algebraic Varieties

Author: Janos Kollar

Publisher: Springer Science & Business Media

Published: 2013-04-09

Total Pages: 330

ISBN-13: 3662032767

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The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.

Néron Models
Néron Models

Author: Siegfried Bosch

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 336

ISBN-13: 3642514383

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Néron models were invented by A. Néron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Néron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Néron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Néron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Néron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between Néron models and the relative Picard functor in the case of Jacobian varieties is explained. The authors helpfully remind the reader of some important standard techniques of algebraic geometry. A special chapter surveys the theory of the Picard functor.