Model theory of fields and compact complex manifolds

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Mathematical logic is the study of the kind of reasoning that mathematicians use. One of the major intellectual achievements of the 20th century was the understanding that this metamathematical pursuit has dramatic consequences for mathematics itself. I have in mind here the celebrated work of Goedel which used mathematical logic to expose the inherent and insurmountable limits of mathematics. But another, less widely known, discovery of the 20th century is that the study of mathematical reasoning can also contribute positively to mathematics; it can produce concrete developments in particular areas of core mathematics such as geometry, algebra and number theory. My own expertise, and this proposal, are informed by this latter positive tendency in the application of mathematical logic, which finds its home in a branch called model theory. Roughly speaking, the model theoretic approach to mathematics is to fix beforehand the particular mathematical structure suited to the area of mathematics under consideration, and then to study those sets that can be defined using formal expressions that refer only to this structure and that are bound syntactically by the rules of first-order logic. These self-imposed restraints on the syntax, which is characteristic of logic, is the source of model theory's effectiveness. The particular applications of model theory to mathematics that I propose to pursue have to do with geometry. Complex manifolds are mathematical abstractions of the idea of shape and form, and their study is an active area of mainstream contemporary mathematics. The involvement of model theory is relatively new, approximately fifteen years old, and has been at the centre of my work. Not only does model theory contribute to the geometry of complex manifolds, but in a kind of reverse process, these contributions have themselves inspired developments in the techniques of pure model theory itself. In turn, these techniques can be re-applied to other geometric contexts. The research I am proposing will further this special role that model theory plays of recognising, formalising and facilitating analogies between different geometric contexts.