Lectures on Arakelov Geometry

Author: C. Soulé
Publisher: Cambridge University Press
ISBN: 9780521477093
Format: PDF, Kindle
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Arakelov theory is a new geometric approach to diophantine equations. It combines algebraic geometry, in the sense of Grothendieck, with refined analytic tools such as currents on complex manifolds and the spectrum of Laplace operators. It has been used by Faltings and Vojta in their proofs of outstanding conjectures in diophantine geometry. This account presents the work of Gillet and Soulé, extending Arakelov geometry to higher dimensions. It includes a proof of Serre's conjecture on intersection multiplicities and an arithmetic Riemann-Roch theorem. To aid number theorists, background material on differential geometry is described, but techniques from algebra and analysis are covered as well. Several open problems and research themes are also mentioned.

Algebraic Geometry and Number Theory

Author: Hussein Mourtada
Publisher: Birkhäuser
ISBN: 331947779X
Format: PDF, Docs
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This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.

Geometry Analysis and Probability

Author: Jean-Benoît Bost
Publisher: Birkhäuser
ISBN: 3319496387
Format: PDF, ePub, Mobi
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This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry. Contributions by: K. Behrend N. Bergeron S. K. Donaldson J. Dubédat B. Duplantier G. Faltings E. Getzler G. Kings R. Mazzeo J. Millson C. Moeglin W. Müller R. Rhodes D. Rössler S. Sheffield A. Teleman G. Tian K-I. Yoshikawa H. Weiss W. Werner The collection is a valuable resource for graduate students and researchers in these fields.

Diophantine Approximation and Abelian Varieties

Author: Bas Edixhoven
Publisher: Springer Science & Business Media
ISBN: 9783540575283
Format: PDF, Kindle
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The 13 chapters of this book centre around the proof of Theorem 1 of Faltings' paper "Diophantine approximation on abelian varieties", Ann. Math.133 (1991) and together give an approach to the proof that is accessible to Ph.D-level students in number theory and algebraic geometry. Each chapter is based on an instructional lecture given by its author ata special conference for graduate students, on the topic of Faltings' paper.

Science and Its Times 1950 present

Author: Neil Schlager
Publisher: Gale / Cengage Learning
ISBN: 9780787639396
Format: PDF, ePub, Docs
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"This series discusses how the major fields of science developed during specific time periods. Each volume focuses on a range of years and includes developments in exploration, life sciences, mathematics, physical sciences, and technology. When the series is completed, the seven volumes will cover 2000 B.C. to the present."--"Outstanding Reference Sources," American Libraries, May 2001.