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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: (DLC) 2016056302 (OCoLC)965754191 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Alexandru Buium; American Mathematical Society, |

ISBN: | 9781470440893 147044089X |

OCLC Number: | 1000357453 |

Description: | 1 online resource. |

Contents: | Cover; Title page; Contents; Preface; Acknowledgments; Introduction; 0.1. Outline of the theory; 0.2. Comparison with other theories; Chapter 1. Algebraic background; 1.1. Algebra; 1.2. Algebraic geometry; 1.3. Superalgebra; Chapter 2. Classical differential geometry revisited; 2.1. Connections in principal bundles and curvature; 2.2. Lie algebra and classical groups; 2.3. Involutions and symmetric spaces; 2.4. Logarithmic derivative and differential Galois groups; 2.5. Chern connections: the symmetric/anti-symmetric case; 2.6. Chern connections: the hermitian case 2.7. Levi-Cività connection and Fedosov connection2.8. Locally symmetric connections; 2.9. Ehresmann connections attached to inner involutions; 2.10. Connections in vector bundles; 2.11. Lax connections; 2.12. Hamiltonian connections; 2.13. Cartan connection; 2.14. Weierstrass and Riccati connections; 2.15. Differential groups: Cassidy and Painlevé; Chapter 3. Arithmetic differential geometry: generalities; 3.1. Global connections and their curvature; 3.2. Adelic connections; 3.3. Semiglobal connections and their curvature; Galois connections 3.4. Curvature via analytic continuation between primes3.5. Curvature via algebraization by correspondences; 3.6. Arithmetic jet spaces and the Cartan connection; 3.7. Arithmetic Lie algebras and arithmetic logarithmic derivative; 3.8. Compatibility with translations and involutions; 3.9. Arithmetic Lie brackets and exponential; 3.10. Hamiltonian formalism and Painlevé; 3.11. -adic connections on curves: Weierstrass and Riccati; Chapter 4. Arithmetic differential geometry: the case of _{ }; 4.1. Arithmetic logarithmic derivative and Ehresmann connections 4.2. Existence of Chern connections4.3. Existence of Levi-Cività connections; 4.4. Existence/non-existence of Fedosov connections; 4.5. Existence/non-existence of Lax-type connections; 4.6. Existence of special linear connections; 4.7. Existence of Euler connections; 4.8. Curvature formalism and gauge action on _{ }; 4.9. Non-existence of classical -cocycles on _{ }; 4.10. Non-existence of -subgroups of simple groups; 4.11. Non-existence of invariant adelic connections on _{ }; Chapter 5. Curvature and Galois groups of Ehresmann connections 5.1. Gauge and curvature formulas5.2. Existence, uniqueness, and rationality of solutions; 5.3. Galois groups: generalities; 5.4. Galois groups: the generic case; Chapter 6. Curvature of Chern connections; 6.1. Analytic continuation along tori; 6.2. Non-vanishing/vanishing of curvature via analytic continuation; 6.3. Convergence estimates; 6.4. The cases =1 and =1; 6.5. Non-vanishing/vanishing of curvature via correspondences; Chapter 7. Curvature of Levi-Cività connections; 7.1. The case =1: non-vanishing of curvature mod ; 7.2. Analytic continuation along the identity |

Series Title: | Mathematical surveys and monographs, no. 222. |

Responsibility: | Alexandru Buium. |

### Abstract:

Introduces and develops an arithmetic analogue of classical differential geometry. 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.
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