Description: O-minimality was isolated (by van den Dries) and developed (by Pillay and Steinhorn) in the 80’s as a fundamental concept in the model theoretic approach to the study of ordered structures. It provides a framework for understanding the geometric properties of sets definable in real closed fields, in part by adapting many results from semi- algebraic geometry to the o-minimal context.

Over the last two decades, there has been a surge of interest around the interaction between o-minimality and diophantine geometry/number theory. Indeed, using his work with Wilkie on the distribution of rational points on o-minimal definable subsets of the reals, Pila gave an unconditional proof of the so-called “modular” Andr ́e-Oort conjecture. This has been vastly generalized to various arithmetic settings culminating in the proof of the full Andr ́e-Oort conjecture by Pila, Shankar and Tsimerman.

This course will consist of two parts. In the first, we will introduce the basics of o-minimality and study the main consequences such as the Cell Decomposition Theorem. Most of the course will be devoted to this part. In the second part, we will explore the Pila-Wilkie theorem and aim to explain how it has been used in the above applications.

Prerequisites: Only some basic knowledge of first-order logic, model theory, and abstract algebra is assumed.

Course text: Lecture notes will be provided.

Other relevant texts:

1. L. van den Dries, Tame Topology and O-Minimal Structures, London Math. Soc. Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.

2. D. Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, 2000.

3. J. Pila and A. Wilkie, The rational points of a definable set, Duke Math. J. 133, no. 3, pg. 591-616, 2006.

4. N. Bhardwaj and L. van den Dries, On the Pila-Wilkie theorem, Expo. Math., vol 40, no. 3, pg. 495-542, 2022.
   
   

Homework: There will be no homework. A list of exercises will be provided.

Accommodations: Please see the full syllabus here.