## Departmental PhD Thesis Exam – Abhishek Oswal

Wednesday, June 3, 2020
2:00 p.m.

PhD Candidate:  Abhishek Oswal
Supervisor:   Jacob Tsimerman
Thesis title: A non-archimedean definable Chow theorem

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O-minimality has had some striking applications to number theory. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame’ property is the following surprising generalization of Chow’s theorem proved by Peterzil and Starchenko – A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this thesis, we explore a non-archimedean analogue of an o-minimal structure and prove a version of the definable Chow theorem in this context.

A copy of the thesis can be found here: thesis-draft-v4