The proposed research concerns the use of ideas from mathematical logic to enhance the understanding of some central objects in mathematics: symmetries, functions, and differential equations. The part of mathematical logic which is relevant to the proposed research is model theory, which is about the ways in which mathematical objects or classes of objects are defined linguistically. Although mathematical logic has traditional connections with the philosophy and foundations of mathematics, there have been many recent applications to core areas of mathematics, and the proposed research is in the latter tradition. Among key problems to be studied are: (i) classifying the ways in which a "group" (collection of symmetries) can act as symmetries of a nice space, sometimes under some model theoretic assumptions, and (ii) what are the equations satisfied by some very special functions, such as exponential functions.
Here are some more details, written for mathematicians. The proposed research has two main parts. A central theme of the first part is the classification of groups definable in models of a first order theory without the independence property. This belongs to the "tame" model theory of groups, generalizing "stable group theory" which itself generalizes part of the theory of algebraic groups. Although the motivation belongs largely to "pure" model theory, this part of the proposal touches also on topological dynamics and Lie groups. For example we expect to produce "new" invariants of a discrete group, with many questions raised. The second part of the proposal deals with applications of model theory to diophantine geometry over function fields, building on earlier work of the PI and others. One problem is the extension of Ax-Schanuel-type theorems, on exponentiation and transcendence, to families of semiabelian varieties. Another problem is to obtain a "transparent" account of the positive characteristic Mordell-Lang conjecture (due originally to Hrushovski).
