Student Algebraic Geometry Seminar: A conjecture in Diophantine geometry, via model theory
Seminar | May 1 | 4-5 p.m. | 891 Evans Hall
Michael Wan, UC Berkeley
In 2011, Jonathan Pila published an unconditional proof of the André-Oort conjecture in Diophantine geometry, in the Annals. A simplified, vague form of the statement is as follows: if X is an irreducible affine complex variety, and X contains a Zariski dense set of “special points”, then X is a “special variety”.
Pila’s proof uses the theory of o-minimality from model theory, a branch of mathematical logic. In particular, it invokes the Pila-Wilkie counting theorem, which gives a bound on the number of rational points on an o-minimal definable set. Similar techniques have yielded a number of other recent results in Diophantine geometry.
We will focus on giving precise statements, and try to roughly outline how the Pila-Wilkie counting theorem helps prove the above version of the André-Oort conjecture. We do not assume knowledge of model theory, and indeed, we will content ourselves with showing how it is used, rather than how it works. We also do not assume (nor have) any knowledge of number theory, and the algebraic geometry involved is purely classical.
Our presentation will draw heavily from Thomas Scanlon’s Séminaire Bourbaki survey, “A proof of the André-Oort conjecture via mathematical logic.”