Zariski geometries: geometry from the logician's point of view
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge University Press
2010
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society lecture note series
360 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XI, 212 S. |
| ISBN: | 9780521735605 0521735602 |
Staff View
MARC
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| 245 | 1 | 0 | |a Zariski geometries |b geometry from the logician's point of view |c Boris Zilber |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2010 | |
| 300 | |a XI, 212 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a London Mathematical Society lecture note series |v 360 | |
| 500 | |a Includes bibliographical references and index | ||
| 600 | 1 | 4 | |a Zariski, Oscar <1899-1986> |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 4 | |a Zariski surfaces | |
| 650 | 0 | 7 | |a Zariski-Topologie |0 (DE-588)4190537-4 |2 gnd |9 rswk-swf |
| 653 | |a Zariski, Oscar, 1899-1986 | ||
| 653 | |a Zariski surfaces | ||
| 653 | |a Geometry, Algebraic | ||
| 689 | 0 | 0 | |a Zariski-Topologie |0 (DE-588)4190537-4 |D s |
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Record in the Search Index
| _version_ | 1804141068626690048 |
|---|---|
| adam_text | Contents
Acknowledgments page
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-smoothness
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 TV-cover of the
affine line
109
5.2.2
Semi-definable functions on P^
109
5.2.3
Space of semi-definable functions 111
5.2.4
Representation of
Λ
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 proj ective 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.I 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.I 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
Hrushovski s construction of new stable structures
202
B.2.3 Pseudo-exponentiation
205
References
207
Index
210
Zariski Geometries
*
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Boris
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This book presents methods and results from the theory of Zanski structures
г
and discusses their applications in geometry as well as various other
mathematical fields, Its logical approach helps us understand why algebraic
geometry is so fundamental iJiroimnout mathematics and why the extension to
non-commutative geometry; which has been forced by recent developments in
quantum physics, is both natural and necessary.
Вееіппіпц
with a crash course in model theory, this book will suit
noi
only
model theorists,
buí
also readers with a more classical geometric background.
|
| any_adam_object | 1 |
| author | Zilber, Boris |
| author_GND | (DE-588)14102545X |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 320 SK 300 |
| classification_tum | MAT 146f |
| ctrlnum | (OCoLC)465366255 (DE-599)BSZ318777185 |
| dewey-full | 516.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.35 |
| dewey-search | 516.35 |
| dewey-sort | 3516.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV036040053 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:10:00Z |
| institution | BVB |
| isbn | 9780521735605 0521735602 |
| language | English |
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| physical | XI, 212 S. |
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| spelling | Zilber, Boris Verfasser (DE-588)14102545X aut Zariski geometries geometry from the logician's point of view Boris Zilber 1. publ. Cambridge [u.a.] Cambridge University Press 2010 XI, 212 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 360 Includes bibliographical references and index Zariski, Oscar <1899-1986> Geometry, Algebraic Zariski surfaces Zariski-Topologie (DE-588)4190537-4 gnd rswk-swf Zariski, Oscar, 1899-1986 Zariski-Topologie (DE-588)4190537-4 s DE-604 London Mathematical Society lecture note series 360 (DE-604)BV000000130 360 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018931954&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018931954&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Zilber, Boris Zariski geometries geometry from the logician's point of view London Mathematical Society lecture note series Zariski, Oscar <1899-1986> Geometry, Algebraic Zariski surfaces Zariski-Topologie (DE-588)4190537-4 gnd |
| subject_GND | (DE-588)4190537-4 |
| title | Zariski geometries geometry from the logician's point of view |
| title_auth | Zariski geometries geometry from the logician's point of view |
| title_exact_search | Zariski geometries geometry from the logician's point of view |
| title_full | Zariski geometries geometry from the logician's point of view Boris Zilber |
| title_fullStr | Zariski geometries geometry from the logician's point of view Boris Zilber |
| title_full_unstemmed | Zariski geometries geometry from the logician's point of view Boris Zilber |
| title_short | Zariski geometries |
| title_sort | zariski geometries geometry from the logician s point of view |
| title_sub | geometry from the logician's point of view |
| topic | Zariski, Oscar <1899-1986> Geometry, Algebraic Zariski surfaces Zariski-Topologie (DE-588)4190537-4 gnd |
| topic_facet | Zariski, Oscar <1899-1986> Geometry, Algebraic Zariski surfaces Zariski-Topologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018931954&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018931954&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
| work_keys_str_mv | AT zilberboris zariskigeometriesgeometryfromthelogicianspointofview |