Foundations of arithmetic differential geometry
 Responsibility
 Alexandru Buium.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2017]
 Physical description
 x, 344 pages ; 27 cm.
 Series
 Mathematical surveys and monographs ; no. 222.
Access
Available online
Science Library (Li and Ma)
Stacks
Call number  Status 

QA3 .A4 V.222  Unknown 
More options
Creators/Contributors
 Author/Creator
 Buium, Alexandru, 1955 author.
 Contributor
 American Mathematical Society, issuing body, publisher.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 339342) and index.
 Contents

 Algebraic backgroundClassical differential geometry revisitedArithmetic differential geometry: GeneralitiesArithmetic differential geometry: The case of $GL_n$Curvature and Galois groups of Ehresmann connectionsCurvature of Chern connectionsCurvature of LeviCivita connectionsCurvature of Lax connectionsOpen problemsBibliographyIndex.
 (source: Nielsen Book Data)9781470436230 20170731
 Publisher's Summary
 The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices.One of the main conclusions of the theory is that the spectrum of the integers is ``intrinsically curved''; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
(source: Nielsen Book Data)9781470436230 20170731
Subjects
 Subject
 Geometry, Differential.
 Geometry, Differential.
 Number theory > Forms and linear algebraic groups > Classical groups.
 Number theory > Discontinuous groups and automorphic forms > $p$adic theory, local fields.
 Algebraic geometry > Arithmetic problems. Diophantine geometry > Local ground fields.
 Differential geometry > Global differential geometry > Methods of Riemannian geometry, including PDE methods; curvature restrictions.
 Geometry, Differential.
 Number theory > Forms and linear algebraic groups > Classical groups.
 Number theory > Discontinuous groups and automorphic forms > $p$adic theory, local fields.
 Geometry, Algebraic > Arithmetic problems. Diophantine geometry > Local ground fields.
 Geometry, Differential > Global differential geometry > Methods of Riemannian geometry, including PDE methods; curvature restrictions.
Bibliographic information
 Publication date
 2017
 Series
 Mathematical surveys and monographs ; volume 222
 ISBN
 9781470436230 hardcover
 147043623X hardcover