Zariski Geometries : Geometry from the Logician's Point of View
Boris. Zilber
Bok Engelsk 2010 · Electronic books.
| Annen tittel | |
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| Utgitt | Cambridge : Cambridge University Press , 2010
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| Omfang | 1 online resource (226 p.)
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| Opplysninger | Description based upon print version of record.. - Title; Copyright; Dedication; Contents; Acknowledgments; 1 Introduction; 1.1 Introduction; 1.2 About model theory; 2 Topological structures; 2.1 Basic notions; 2.2 Specialisations; 2.2.1 Universal specialisations; 2.2.2 Infinitesimal neighbourhoods; 2.2.3 Continuous and differentiable function; 3 Noetherian Zariski structures; 3.1 Topological structures with good dimension notion; 3.1.1 Good dimension; 3.1.2 Zariski structures; 3.2 Model theory of Zariski structures; 3.2.1 Elimination of quantifiers; 3.2.2 Morley rank; 3.3 One-dimensional case; 3.4 Basic examples. - 3.4.1 Algebraic varieties and orbifolds over algebraically closed fields3.4.2 Compact complex manifolds; 3.4.3 Proper varieties of rigid analytic geometry; 3.4.4 Zariski structures living in differentially closed fields; 3.5 Further geometric notions; 3.5.1 Pre-smoothness; 3.5.2 Coverings in structures with dimension; 3.5.3 Elementary extensions of Zariski structures; 3.6 Non-standard analysis; 3.6.1 Coverings in pre-smooth structures; 3.6.2 Multiplicities; 3.6.3 Elements of intersection theory; 3.6.4 Local isomorphisms; 3.7 Getting new Zariski sets; 3.8 Curves and their branches. - 4 Classification results4.1 Getting a group; 4.1.1 Composing branches of curves; 4.1.2 Pre-group of jets; 4.2 Getting a field; 4.3 Projective spaces over a Z-field; 4.3.1 Projective spaces as Zariski structures; 4.3.2 Completeness; 4.3.3 Intersection theory in projective spaces; 4.3.4 Generalised Bezout and Chow theorems; 4.4 The classification theorem; 4.4.1 Main theorem; 4.4.2 Meromorphic functions on a Zariski set; 4.4.3 Simple Zariski groups are algebraic; 5 Non-classical Zariski geometries; 5.1 Non-algebraic Zariski geometries; 5.2 Case study; 5.2.1 The N-cover of the affine line. - 5.2.2 Semi-definable functions on PN5.2.3 Space of semi-definable functions; 5.2.4 Representation of A; 5.2.5 Metric limit; 5.3 From quantum algebras to Zariski structures; 5.3.1 Algebras at roots of unity; 5.3.2 Examples; 5.3.3 Definable sets and Zariski properties; 6 Analytic Zariski geometries; 6.1 Definition and basic properties; 6.1.1 Closed and projective sets; 6.1.2 Analytic subsets; 6.2 Compact analytic Zariski structures; 6.3 Model theory of analytic Zariski structures; 6.4 Specialisations in analytic Zariski structures; 6.5 Examples; 6.5.1 Covers of algebraic varieties. - 6.5.2 Hard examplesA Basic model theory; A.1 Languages and structures; A.2 Compactness theorem; A.3 Existentially closed structures; A.4 Complete and categorical theories; A.4.1 Types in complete theories; A.4.2 Spaces of types and saturated models; A.4.3 Categoricity in uncountable powers; B Elements of geometric stability theory; B.1 Algebraic closure in abstract structures; B.1.1 Pre-geometry and geometry of a minimal structure; B.1.2 Dimension notion in strongly minimal structures; B.1.3 Macro- and micro-geometries on a strongly minimal structure; B.2 Trichotomy conjecture. - B.2.1 Trichotomy conjecture. - Methods and results from the theory of Zariski structures, and their applications in geometry.
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| Emner | |
| Sjanger | |
| Dewey | |
| ISBN | 9780521735605
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