Thomas Scanlon (University of California)
O-minimal methods in geometry
O-minimality is an apparently weak tameness hypothesis on the class of sets definable in a given ordered structure, namely that the definable subsets of the line are finite unions of points and intervals, which implies strong regularity properties for all of the definable sets in any dimension. While o-minimality concerns (possible generalizations of) real semi-algebraic and semi-analytic geometry, it has consequences for complex analytic geometry through the work of Peterzil and Starchenko on o-minimal complex analysis and for Diophantine geometry through work of Pila and Wilkie on counting rational points.
I will survey some of the history of this subject pointing out the key papers and will close with discussions about uses of o-minimality in the study of algebraic differential equations and functional transcendence.