Model theory with applications to algebra, geometry and number theory

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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 subject 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 study have to do with geometry and algebra. Algebraic differential equations are at once a classical and currently active subject in mathematics arising from the study of the natural world. Often differential equations are considered part of applied mathematics. But there is a beautiful pure-mathematical aspect to the subject, and it is here that my work lies. In particular, I approach such equations by considering the solutions space as a geometric object. Not only does model theory contribute to differential-algebraic geometry, but in a kind of reverse process, these contributions themselves inspire the development of new techniques in pure model theory itself. In turn, these techniques can be re-applied to other geometric contexts. The research I am proposing will further this dynamical two-way interaction of model theory with geometry and algebra. The long term goal of the proposed research is to advance the field of model theory, through the study of applications to geometry and algebra. This will be achieved through the pursuit of the following complementary research themes: 1) the specific context of Differential-algebraic Geometry, 2) the pure model theory developments in Geometric Stability Theory inspired by (1) as well as other settings, and finally 3) the Dixmier-Moeglin Equivalence as a particular programme of applications illustrating the power of the model-theoretic approach.