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OverviewThe 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. Full Product DetailsAuthor: Alexandru BuiumPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.792kg ISBN: 9781470436230ISBN 10: 147043623 Pages: 336 Publication Date: 30 May 2017 Audience: Professional and scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsAlgebraic background Classical differential geometry revisited Arithmetic differential geometry: Generalities Arithmetic differential geometry: The case of $GL_n$ Curvature and Galois groups of Ehresmann connections Curvature of Chern connections Curvature of Levi-Civita connections Curvature of Lax connections Open problems Bibliography Index.ReviewsAuthor InformationAlexandru Buium, University of New Mexico, Albuquerque, NM. Tab Content 6Author Website:Countries AvailableAll regions |
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