Sat Jan 29, 2011
60 Evans Hall
Thomas Scanlon (Professor of Mathematics, University of California, Berkeley)
Number-theoretic Consequences of the Model Theory of Real Geometry
We say that a first-order structure (R, <, …) in some language expanding the language of orders is o-minimal if < linearly orders R and every definable subset of R is a finite union of points and intervals. Tarski’s quantifier elimination theorem for the real field implies that the real numbers considered as an ordered ring is o-minimal and subsequent work by various authors has established the o-minimality of more sophisticated structures on the real numbers. The definable sets in an o-minimal structure (in any number of variables) admit a very tame geometric structure theory, very much at odds with the Gödel phenomena in arithmetic. Nevertheless, Pila and Wilkie bridged the divide between geometry and arithmetic by establishing bounds on the number of rational points in sets definable in o-minimal expansions of the real numbers. In recent work, Pila employed this counting theorem to prove the celebrated André-Oort conjecture on algebraic relations on special points of modular curves.
In this talk, I will describe the Pila-Wilkie counting theorem and some of my own work relating arithmetic questions about algebraic dynamical systems to o-minimality and the Pila-Wilkie bounds.
The biweekly LOGIC TEA will be held in the Alfred Tarski Room (727 Evans Hall) immediately following the Colloquium (with support from the Graduate Assembly).