2nd Semester 2025/26: O-minimality and Tame Topology

Instructors

Alexi Block Gorman

ECTS

6

Description

This course is a continuation of the MoL model theory course, but does not require an extensive mathematical background. In many ways, o-minimality is to model theory what algebraic geometry is algebra. The very simple definition of an o-minimal structure--one in which all one-dimensional definable sets are a finite union of points and intervals--belies the powerful tools that o-minimality places at our disposal. Indeed, o-minimal expansions of the field of real numbers is by many regarded as the "correct" setting in which to do real analytic geometry. In this project, we will see why that is, and we will also examine which parts of complex analytic geometry we can use o-minimality for as well. In addition, we will look at the far-ranging applications of o-minimality, which vary widely from point-counting theorems in number theory to computations in Hodge theory to probably approximately correct (PAC) learnable classes in machine learning.

Organisation

The first two weeks will consist of readings, presentations on the basic material, and exercises. The second two weeks will consist of student presentations.

Prerequisites

Model Theory

Assessment

Student presentations and exercises.

References

Tame Topology and O-Minimal Structures, by Lou van den Dries