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London Mathematical Society and EPSRC Short Instructional Course - Model Theory
University of Leeds
18-23 July 2010

Organiser: Dugald Macpherson

Lecture Course II Introduction to o-minimality with applications (Marcus Tressl, Manchester)

The subject of o-minimality was created in the late 1970s inside mathematical logic (and is still a major research topic there) as a framework for `tame' mathematics. Here, tameness means to work rigorously from the beginning with mathematical objects that are tractable in applications, rather than dealing with theoretical pathologies in the classical formulation. The concept was rapidly absorbed into real analytic geometry, where people have been looking for such a tame context since the beginning of the last century. Tameness as presented in o-minimality is widely considered as a realisation of Grothendieck's demand on how to develop mathematics (see his `Esquisse d'un programme').

The course will give an introduction to o-minimality including the most fundamental theorems on o-minimal structures, like cell decomposition, ordinary dimension theory, Euler characteristic, Vapnik-Chervonenkis dimension (with applications to learnability of neural networks). The course will describe three important o-minimal structures, the real field, the real field expanded by analytic functions restricted to bounded sets and the real field together with the global exponential function. We will also demonstrate the key role of o-minimality in two areas: (i) decidability questions of structures on the real field, (ii) real algebraic and analytic geometry.

Last updated: Wed 30 Jan 2008