O-minimality is a property of ordered structures which yields results generalizing the classical finiteness theorems long known to hold for semialgebraic and subanalytic sets, such as the existence of cell decompositions and Whitney stratifications. This leads to a development of a kind of "tame topology" (envisaged by Grothendieck). Although originating in model theory, over the last twenty years, the notion of an o-minimal structure has proven to be increasingly useful in the fields of real algebraic and real analytic geometry.  The general theory has even had applications to subjects as varied as Lie theory, economics, and neural networks.

This course will be split into two parts. In the first half, we will introduce the o-minimality axiom and develop its main consequences. We will discuss basic facts about o-minimal structures: their geometric, model-theoretic, and algebraic (valuation-theoretic) properties. The second half of the course will be designed according to the interests of the participants. Possibilities include the discussion of some of the various new and mathematically interesting examples of o-minimal structures, and methods to construct them, which have emerged in recent years; or a further development of the valuation theory of o-minimial structures, with applications to limit sets and tropical geometry.

Prerequisities

Some basic knowledge of first-order logic, model theory, and abstract algebra should be sufficient. No knowledge of real algebraic and analytic geometry will be assumed.

There will be an occasional problem set assigned on a semi-regular basis, handed out in class, and also posted on this website.