Foundations of arithmetic differential geometry / Alexandru Buium.

Format:

Book

Author/Creator:

Buium, Alexandru, 1955- author.

Series:

Mathematical surveys and monographs ; no. 222.

Mathematical surveys and monographs ; volume 222

Language:

English

Subjects (All):

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.

Local Subjects:

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.

Physical Description:

x, 344 pages ; 27 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2017]

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."--Page 4 of cover.

Contents:

Introduction

Algebraic background

Classical differential geometry revisited

Arithmetic differential geometry : generalities

Arithmetic differential geometry : the case of GLn

Curvature and Galois groups of Ehresmann connections

Curvature of Chern connections

Curvature of Levi-Civitáa connections

Curvature of Lax connections

Open problems.

Notes:

Includes bibliographical references (pages 339-342) and index.

ISBN:

9781470436230

147043623X

OCLC:

965754191