کتابخانه دیجیتال فوركيا

کتابخانه دیجیتال فوركيا

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Foundations of Arithmetic Differential Geometry


Foundations of Arithmetic Differential Geometry

Title: Foundations of Arithmetic Differential Geometry | Author(s): Alexandru Buium | Publisher: American Mathematical Society | Year: 2017 | Language: English | Pages: 344 | Size: 4 MB | Extension: pdf

 

Alexandru Buium: University of New Mexico, Albuquerque, NM

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.

 

Readership

Graduate students and researchers interested in algebraic geometry, number theory, and algebraic groups.

 

 

Table of Contents

 
 
  • Cover 1
  • Title page 4
  • Contents 6
  • Preface 8
    • Acknowledgments 11
  • Introduction 12
    • 0.1. Outline of the theory 12
    • 0.2. Comparison with other theories 33
  • Chapter 1. Algebraic background 38
    • 1.1. Algebra 38
    • 1.2. Algebraic geometry 43
    • 1.3. Superalgebra 44
  • Chapter 2. Classical differential geometry revisited 50
    • 2.1. Connections in principal bundles and curvature 50
    • 2.2. Lie algebra and classical groups 64
    • 2.3. Involutions and symmetric spaces 69
    • 2.4. Logarithmic derivative and differential Galois groups 74
    • 2.5. Chern connections: the symmetric/anti-symmetric case 75
    • 2.6. Chern connections: the hermitian case 81
    • 2.7. Levi-Cività connection and Fedosov connection 83
    • 2.8. Locally symmetric connections 89
    • 2.9. Ehresmann connections attached to inner involutions 90
    • 2.10. Connections in vector bundles 91
    • 2.11. Lax connections 93
    • 2.12. Hamiltonian connections 96
    • 2.13. Cartan connection 103
    • 2.14. Weierstrass and Riccati connections 105
    • 2.15. Differential groups: Cassidy and Painlevé 106
  • Chapter 3. Arithmetic differential geometry: generalities 110
    • 3.1. Global connections and their curvature 110
    • 3.2. Adelic connections 123
    • 3.3. Semiglobal connections and their curvature; Galois connections 126
    • 3.4. Curvature via analytic continuation between primes 130
    • 3.5. Curvature via algebraization by correspondences 136
    • 3.6. Arithmetic jet spaces and the Cartan connection 149
    • 3.7. Arithmetic Lie algebras and arithmetic logarithmic derivative 158
    • 3.8. Compatibility with translations and involutions 163
    • 3.9. Arithmetic Lie brackets and exponential 170
    • 3.10. Hamiltonian formalism and Painlevé 172
    • ...


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