Originating in model theory, o-minimality is a tameness property of real sets which enjoys important finiteness properties. O-minimal structures have recently found a number of important applications to algebraic and arithmetic geometry, including functional transcendence, the Andre-Oort conjecture, and Hodge theory. The aim of this summer school is to provide an introduction to these ideas for an audience of non-experts.

### Date and location

The school takes place from September 09 to September 13, 2019 at the University of Freiburg, Germany

### Lecture Series

We plan for three lecture series on o-minimal Structures, delivered by Benjamin Bakker, Yohan Brunebarbe and Bruno Klingler.

• o-minimal Structures and Algebraization
• The general tameness properties of o-minimal structures
• o-minimality of $R_{an}$ and $R_{an, exp}$
• Definable analytic spaces and definable coherent sheaves
• The Closure Theorem, o-minimal Chow, and o-minimal GAGA
• Functional Transcendence and Diophantine Geometry
• Functional transcendence.
• Illustration for tori, Abelian varieties, Shimura varieties, and general variations of Hodge structures
• The Pila-Wilkie Theorem
• The Ax-Schanuel Conjecture
• Applications to Hodge Theory
• Variations of Hodge structures and period maps
• Definability of uniformizations of Shimura varieties and period maps
• Applications to the algebraicity of Hodge loci and the Griffiths Conjecture

### Additional Lectures

• “Introduction to o-minimality” by Amador Martin-Pizarro

### Sponsor

Funding is provided by Freiburg’s Graduiertenkolleg GRK1821 “Cohomological Methods in Geometry”.