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Foundations of Arithmetic Differential Geometry

Alexandru Buium
Format
Book
Published
Providence, Rhode Island : American Mathematical Society, [2017]
Language
English
Series
Mathematical Surveys and Monographs
ISBN
9781470436230, 147043623X
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-Cività connections
  • Curvature of Lax connections
  • Open problems.
Description
x, 344 pages ; 27 cm.
Notes
Includes bibliographical references (pages 339-342) and index.
Series Statement
Mathematical surveys and monographs ; volume 222
Mathematical surveys and monographs ; no. 222
Technical Details
  • Access in Virgo Classic
  • Staff View

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    a| DLC b| eng e| rda c| DLC d| OCLCF d| OCLCQ d| OCLCO d| YDX d| BTCTA d| BUF d| KSU d| GZN d| YDX d| OCLCO d| OBE d| NRC
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    a| Buium, Alexandru, d| 1955- e| author.
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    a| Foundations of arithmetic differential geometry / c| Alexandru Buium.
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    a| Providence, Rhode Island : b| American Mathematical Society, c| [2017]
    300
      
      
    a| x, 344 pages ; c| 27 cm.
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    a| text b| txt 2| rdacontent
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    a| unmediated b| n 2| rdamedia
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    a| volume b| nc 2| rdacarrier
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    a| Mathematical surveys and monographs ; v| volume 222
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    a| Includes bibliographical references (pages 339-342) and index.
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    0
      
    a| 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-Cività connections -- Curvature of Lax connections -- Open problems.
    520
      
      
    a| "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.
    650
      
    0
    a| Geometry, Differential.
    650
      
    7
    a| Geometry, Differential. 2| fast 0| (OCoLC)fst00940919
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    7
    a| Number theory x| Forms and linear algebraic groups x| Classical groups. 2| msc
    650
      
    7
    a| Number theory x| Discontinuous groups and automorphic forms x| $p$-adic theory, local fields. 2| msc
    650
      
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    a| Algebraic geometry x| Arithmetic problems. Diophantine geometry x| Local ground fields. 2| msc
    650
      
    7
    a| Differential geometry x| Global differential geometry x| Methods of Riemannian geometry, including PDE methods; curvature restrictions. 2| msc
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    2
      
    a| American Mathematical Society, e| issuing body, e| publisher.
    830
      
    0
    a| Mathematical surveys and monographs ; v| no. 222.
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    a| Z0 b| VA@
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    999
      
      
    a| QA641 .B774 2017 w| LC i| X031815400 l| STACKS m| MATH t| BOOK
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