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O-Minimality and Diophantine Applications
Last Updated: 2026-02-05 16:02:14
Abstract
O-minimal structures provide a framework for "tame topology", as envisioned for instance by Grothendieck. Although motivated by questions in model theory and real algebraic geometry, the notion the o-minimal structures was revealed by Pila, Wilkie, Zannier and others to have remarkable applications to number theory and arithmetic geometry.
Objective
The overall goal of this course is to provide an introduction to o-minimality and the applications of o-minimal structures.
Content
The first part of the course will be devoted to an introduction to model theory as a framework in which to define o-minimal structures. The main result will be the "cell decomposition theorem", which describes the shape of definable subsets of an o-minimal structure. In the second part of the course, we will discuss examples of interesting o-minimal structures, and then consider applications to number theory. These may include Pila-Wilkie counting theorem, or the Pila-Zannier strategy in the contet of the Manin-Mumford conjecture.
Resources
Literature
G. Jones and A. Wilkie: O-minimality and diophantine geometry, Cambridge University Press. L. van den Dries: Tame topology and o-minimal structures, Cambridge University Press. A. Forey: lectures notes on o-minimality and arithmetic applications.
Learning Materials (Links)
- Main link
- Information
General Information
- Language
- English
- Levels
- MSC
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture with exercise | O-Minimality and Diophantine Applications |
|
4 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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