Foundations of Arithmetic Differential Geometry (Mathematical Surveys and Monographs) by Alexandru Buium (PDF)

Description

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.

User’s Reviews

Editorial Reviews: About the Author Alexandru Buium, University of New Mexico, Albuquerque, NM.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Formal similarities between the integers and function fields, or even rings of differentiable functions, are well-known. But this book takes those classical analogies and develops a new and fascinating area of mathematics. Readers should have a solid foundation in algebraic geometry, but if they do, they will find a rich and fascinating vista has opened before them. I was also pleasantly surprised in reading this book to find that it does a good job in laying out the fundamentals of more classical differential geometry (albeit from an algebraic geometric perspective). I am very excited by this book and what it represents to the mathematical community.

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