The model theory of compact complex manifolds and other geometric strucutres

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I am interested in the interaction between mathematical logic and geometry. Mathematical logic, to quote the opening lines of Shoenfield's book by that name, is the study of the type of reasoning done by mathematicians. Geometry is one of the core branches of mathematics itself; it is inspired by (and vastly generalises) the study of objects in three-dimensional space. Somewhat surprisingly, the methods of the meta-mathematical subject of logic have proved to be of substantial geometrical significance. Fundamentally, what logic brings to geometry is a "linguistic" intuition; the logician plays close attention to what can be expressed or defined in a given formal language. Associated to these formal expressions (syntax) are the objects they define (semantics), the "definable sets". Model theory, the particular branch of mathematical logic in which I specialise, could be described as the study of the geometrical properties of definable sets.