Course Identification

Tame geometry is a general name for the study of sets that have "finite complexity" in a suitable sense. Specific examples include the algebraic case (semialgebraic geometry), the analytic case (subanalytic geometry) and the exponential-analytic case. The general subject is formalized within the model-theoretic framework of o-minimal geometry. In the past decade a remarkable link between tame geometry and diophantine geometry ("the geometry of numbers") has been unfolding, leading to the solution of several outstanding conjectures. The goal of this course is to study the basics of tame geometry and then move on to the description of these diophantine applications. 

We will cover the following topics:

• Introduction to first-order logic, model theory, and o-minimality.
• Dimension, cellular decomposition, curve selection.
• The Tarski-Seidenberg theorem and the structure R_alg.
• Gabrielov's theorem of the complement and the structure R_an.
• Wilkie's theorem, the structures R_exp and R_{an,exp}.
• Fewnomial theory and effectivity in tame geometry.
• The Yomdin-Gromov theorem on smooth parametrizations, applications in dynamics.
• The Pila-Wilkie counting theorem.
• Intro to elliptic curves, abelian varieties.
• The Pila-Zannier proof of the Manin-Mumford conjecture.
• Intro to modular curves.
• Pila's proof of the Andre-Oort conjecture for modular curves.
• Functional transcendence results.

Upon completion of the course the student will be able to:

Read research papers in the areas of o-minimality and its diophantine applications and use these ideas in their own research.