UBC Colloquium and PIMS Distinguished Guest Lecturer: Thomas Scanlon

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.