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Title: Non-algebraic Zariski geometries
Author: Sustretov, Dmitry
ISNI:       0000 0004 2744 973X
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2012
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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 I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of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.
Supervisor: Zilber, Boris Sponsor: Hill Foundation
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematical logic and foundations ; Zariski geometries ; interpretability ; algebraic geometry ; model theory