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You searched for subject:(Zariski geometries). Showing records 1 – 2 of 2 total matches.

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University of Oxford

1. Elsner, Bernhard August Maurice. Presmooth geometries.

Degree: PhD, 2014, University of Oxford

This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.

Subjects/Keywords: 516.3; Mathematical logic and foundations; Mathematics; Model Theory; Zariski Geometries; Presmooth Geometries; Trichotomy Theorem; Zilber's Trichotomy

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APA (6th Edition):

Elsner, B. A. M. (2014). Presmooth geometries. (Doctoral Dissertation). University of Oxford. Retrieved from http://ora.ox.ac.uk/objects/uuid:b5d9ccfd-8360-4a2c-ad89-0b4f136c5a96 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.618508

Chicago Manual of Style (16th Edition):

Elsner, Bernhard August Maurice. “Presmooth geometries.” 2014. Doctoral Dissertation, University of Oxford. Accessed June 20, 2019. http://ora.ox.ac.uk/objects/uuid:b5d9ccfd-8360-4a2c-ad89-0b4f136c5a96 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.618508.

MLA Handbook (7th Edition):

Elsner, Bernhard August Maurice. “Presmooth geometries.” 2014. Web. 20 Jun 2019.

Vancouver:

Elsner BAM. Presmooth geometries. [Internet] [Doctoral dissertation]. University of Oxford; 2014. [cited 2019 Jun 20]. Available from: http://ora.ox.ac.uk/objects/uuid:b5d9ccfd-8360-4a2c-ad89-0b4f136c5a96 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.618508.

Council of Science Editors:

Elsner BAM. Presmooth geometries. [Doctoral Dissertation]. University of Oxford; 2014. Available from: http://ora.ox.ac.uk/objects/uuid:b5d9ccfd-8360-4a2c-ad89-0b4f136c5a96 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.618508


University of Oxford

2. Sustretov, Dmitry. Non-algebraic Zariski geometries.

Degree: PhD, 2012, University of Oxford

The thesis deals with definability of certain Zariski geometries, introduced by Zilber, in the theory of algebraically closed fields. I axiomatise a class of structures, called 'abstract linear spaces', which are a common reduct of these Zariski geometries. I then describe what an interpretation of an abstract linear space in an algebraically closed field looks like. I give a new proof that the structure "quantum harmonic oscillator", introduced by Zilber and Solanki, is not interpretable in an algebraically closed field. I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of the group law on a Galois cohomology group H1(k(x), μn). I further consider quantum Zariski geometries introduced by Zilber and give necessary and sufficient conditions that a quantum Zariski geometry be definable in an algebraically closed field. Finally, I take an attempt at extending the results described above to complex-analytic setting. I define what it means for quantum Zariski geometry to have a complex analytic model, an give a necessary and sufficient conditions for a smooth quantum Zariski geometry to have one. I then prove a theorem giving a partial description of an interpretation of an abstract linear space in the structure of compact complex spaces and discuss the difficulties that present themselves when one tries to understand interpretations of abstract linear spaces and quantum Zariski geometries in the compact complex spaces structure.

Subjects/Keywords: 516.3; Mathematical logic and foundations; Zariski geometries; interpretability; algebraic geometry; model theory

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Sustretov, D. (2012). Non-algebraic Zariski geometries. (Doctoral Dissertation). University of Oxford. Retrieved from http://ora.ox.ac.uk/objects/uuid:b67f85d8-6fac-4820-913d-a064d3582412 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.581011

Chicago Manual of Style (16th Edition):

Sustretov, Dmitry. “Non-algebraic Zariski geometries.” 2012. Doctoral Dissertation, University of Oxford. Accessed June 20, 2019. http://ora.ox.ac.uk/objects/uuid:b67f85d8-6fac-4820-913d-a064d3582412 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.581011.

MLA Handbook (7th Edition):

Sustretov, Dmitry. “Non-algebraic Zariski geometries.” 2012. Web. 20 Jun 2019.

Vancouver:

Sustretov D. Non-algebraic Zariski geometries. [Internet] [Doctoral dissertation]. University of Oxford; 2012. [cited 2019 Jun 20]. Available from: http://ora.ox.ac.uk/objects/uuid:b67f85d8-6fac-4820-913d-a064d3582412 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.581011.

Council of Science Editors:

Sustretov D. Non-algebraic Zariski geometries. [Doctoral Dissertation]. University of Oxford; 2012. Available from: http://ora.ox.ac.uk/objects/uuid:b67f85d8-6fac-4820-913d-a064d3582412 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.581011

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