Acknowledgments ................................................ xi
1 Introduction ................................................. 1
1.1 Introduction ............................................ 1
1.2 About model theory ...................................... 7
2 Topological structures ...................................... 12
2.1 Basic notions .......................................... 12
2.2 Specialisations ........................................ 14
2.2.1 Universal specialisations ....................... 17
2.2.2 Infinitesimal neighbourhoods .................... 19
2.2.3 Continuous and differentiable function .......... 22
3 Noetherian Zariski structures ............................... 25
3.1 Topological structures with good dimension notion ...... 25
3.1.1 Good dimension .................................. 25
3.1.2 Zariski structures .............................. 26
3.2 Model theory of Zariski structures ..................... 27
3.2.1 Elimination of quantifiers ...................... 27
3.2.2 Morley rank ..................................... 30
3.3 One-dimensional case ................................... 30
3.4 Basic examples ......................................... 35
3.4.1 Algebraic varieties and orbifolds over
algebraically closed fields ..................... 35
3.4.2 Compact complex manifolds ....................... 36
3.4.3 Proper varieties of rigid analytic geometry ..... 38
3.4.4 Zariski structures living in differentially
closed fields ................................... 39
3.5 Further geometric notions .............................. 40
3.5.1 Pre-smoofhness ................................... 40
3.5.2 Coverings in structures with dimension .......... 43
3.5.3 Elementary extensions of Zariski structures ..... 44
3.6 Non-standard analysis .................................. 50
3.6.1 Coverings in pre-smooth structures .............. 50
3.6.2 Multiplicities .................................. 53
3.6.3 Elements of intersection theory ................. 57
3.6.4 Local isomorphisms .............................. 59
3.7 Getting new Zariski sets ............................... 62
3.8 Curves and their branches .............................. 69
4 Classification results ...................................... 78
4.1 Getting a group ........................................ 78
4.1.1 Composing branches of curves .................... 79
4.1.2 Pre-group of jets ............................... 82
4.2 Getting a field ........................................ 88
4.3 Projective spaces over a Z-field ....................... 93
4.3.1 Projective spaces as Zariski structures ......... 93
4.3.2 Completeness .................................... 94
4.3.3 Intersection theory in projective spaces ........ 95
4.3.4 Generalised Bezout and Chow theorems ............ 97
4.4 The classification theorem ............................ 100
4.4.1 Main theorem ................................... 100
4.4.2 Meromorphic functions on a Zariski set ......... 101
4.4.3 Simple Zariski groups are algebraic ............ 103
5 Non-classical Zariski geometries ........................... 105
5.1 Non-algebraic Zariski geometries ...................... 105
5.2 Case study ............................................ 109
5.2.1 The N-cover of the affine line ................. 109
5.2.2 Semi-definable functions on PN ................. 109
5.2.3 Space of semi-definable functions .............. 111
5.2.4 Representation of  ............................ 111
5.2.5 Metric limit ................................... 115
5.3 From quantum algebras to Zariski structures ........... 120
5.3.1 Algebras at roots of unity ..................... 122
5.3.2 Examples ....................................... 125
5.3.3 Definable sets and Zariski properties .......... 134
6 Analytic Zariski geometries ................................ 137
6.1 Definition and basic properties ....................... 137
6.1.1 Closed and projective sets ..................... 138
6.1.2 Analytic subsets ............................... 139
6.2 Compact analytic Zariski structures ................... 140
6.3 Model theory of analytic Zariski structures ........... 144
6.4 Specialisations in analytic Zariski structures ........ 153
6.5 Examples .............................................. 155
6.5.1 Covers of algebraic varieties .................. 155
6.5.2 Hard examples .................................. 159
A Basic model theory ......................................... 163
A.l Languages and structures .............................. 163
A.2 Compactness theorem ................................... 166
A.3 Existentially closed structures ....................... 170
A.4 Complete and categorical theories ..................... 172
A.4.1 Types in complete theories ..................... 175
A.4.2 Spaces of types and saturated models ........... 177
A.4.3 Categoricity in uncountable powers ............. 182
В Elements of geometric stability theory ..................... 185
B.1 Algebraic closure in abstract structures .............. 185
B.1.1 Pre-geometry and geometry of a minimal
structure ...................................... 186
B.1.2 Dimension notion in strongly minimal
structures ..................................... 189
B.1.3 Macro- and micro-geometries on a strongly
minimal structure .............................. 194
B.2 Trichotomy conjecture ................................. 200
B.2.1 Trichotomy conjecture .......................... 200
B.2.2 Hrashovski's construction of new stable
structures ..................................... 202
B.2.3 Pseudo-exponentiation .......................... 205
References .................................................... 207
Index ......................................................... 210
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