Model theory: from the exponential function to Kummer theory of fields

In recent years it has been understood that a careful model-theoretic analysis of certain classical mathematical theories leads naturally to Schanuel and Mordell-Lang conjectures, Kummer theory, a description of Galois action on roots of unity and Tate modules of elliptic curves, or rather to generalisations of these conjectures. The techniques of model theory show how these conjectures naturally appear in an attempt to find (and prove!) the “right”, “simple”, “geometric” language for those theories, in an appropriate formal formal sense, and to exhibit an isomorphism between the algebraic structures corresponding to those languages. In the talk I will try to demonstrate the main ideas by performing such an analysis of the complex exponential function in the simplest setting; I will avoid model-theoretic terminology and so will say virtually nothing of the Schanuel and Mordell-Lang conjectures.
I will start by an observation that, up to a field automorphism of the complex numbers, there is a unique extension of the multiplicative group of the complex numbers by the infinite cyclic group. The complex exponential function provides such an extension; philosophically, the observation says it is “right”, as is often done, to think of the complex numbers as a field and the exponential function as a group homomorphism into the multiplicative group of a field: the basic properties of this structure determine it uniquely. Often the exponential function is thought of as a universal covering map, and we may think of the exponential function not as the homomorphism as above, but rather come up with another algebraic structure reflecting this topological viewpoint (“fundamental groupoid functor”). I will explain why these two structures are equivalent, some geometry behind their relationship, and how Kummer theory appears in the proofs. I will end by showing that replacing the complex field by an algebraically closed field of high cardinality leads to non-trivial considerations of arithmetical nature, and that considering the Weirstrass function instead of the exponential function leads to (known) conjectures on the image of Galois on the Tate module of an elliptic curve.
The talk is based on ideas of the recent papers of B. Zilber.