## UBC Colloquium and PIMS Distinguished Guest Lecturer: Thomas Scanlon

- Date: 01/06/2012
- Time: 15:00

University of British Columbia

A logician's view of diophantine geometry

Abstract

To a logician diophantine geometry, the study of geometric relations on

points of arithmetic origin, is almost a contradiction in terms since

geometrical reasoning has long been known to be tame and decidable (from

the 1929 work of Tarski on elementary geometry) while by Gödel's

incompleteness theorems arithmetic is known to be infected by inherent

undecidability. However, this tension has not prevented geometers and

number theorists from investigating a subject they know to be hard but

one in which ideas from geometry explain the apparent regularity of the

solutions in rational numbers, roots of unity, or other arithmetically

interesting to certain geometrically constrained systems of equations.

In many important cases, model theorists can explain or prove such

theorems by showing that the structure on the arithmetic points simply

reflects the tameness of the class of definable sets in some structure

intermediate between algebraic geometry and arithmetic.

With this lecture I will broadly survey the project of applying the

model theory of enriched, though still tractable, geometries such as

those coming from differential or difference algebra, to number

theoretic problems such as those around the conjectures of Mordell-Lang

and André-Oort or arising from arithmetic dynamics.