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Foundations of arithmetic differential geometry

Author: Alexandru Buium; American Mathematical Society,
Publisher: Providence, Rhode Island : American Mathematical Society, [2017]
Series: Mathematical surveys and monographs, no. 222.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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  Read more...
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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  Read more...

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