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401-4037-DRL 2 Credits DR D-MATH

O-Minimality and Diophantine Applications

Lecturers & Examiners: Prof. Dr. Emmanuel Kowalski
Only for ZGSM (ETH D-MATH and UZH I-MATH) doctoral students. The latter need to register at myStudies and then send an email to with their name, course number and student ID. Please see
VVZ CR n/a

Last Updated: 2026-02-05 16:02:04

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)

General Information

Language
English
Levels
DR

Examination

Type
ungraded semester performance

Registration & Places

Priority: Registration for the course unit is only possible for the primary target group

Course Components

Type Title Time & Place Hours
lecture with exercise O-Minimality and Diophantine Applications
  • Mon 14:15-16:00 (HG E 41)
  • Thu 14:15-16:00 (HG E 41)
4 h weekly

Offered In