Hall 123

Ax-Grothendieck Theorem - A Model Theoretic Proof.

Venkata Hanumanta Prathamesh

IMSc, Chennai

Ax-Grothendieck Theorem states that if a polynomial map P from a complex vector space to itself is injective, then it is surjective. This result was independently discovered by both Ax and Grothendieck. The proof by Ax uses a simple application of compactness theorem from mathematical logic(/model theory) to generalize the result from finite fields to fields of algebraic characteristic 0. An interesting feature of this proof is that it demonstrates how tools from metamathematical areas like model theory can be used to prove results `in' mathematics, and not just `about' mathematics. In this talk, I will present the proof by Ax.

This talk should be accessible to everyone. No prior knowledge of logic shall be assumed. A brief introduction to model theory and the relevant results shall be introduced in the course of the talk.