September 27, 2023

Thematic Program on Singularity Theory and Geometry
January - June 1997

Coxeter Lecture Series

Mathematics Institute, University of Oxford

A series of three lectures on
O-Minimal Tarski Systems: Theory and Examples

Monday, March 10, 1997 3:30-4:30 p.m.

In this lecture I shall introduce the notion of an o-minimal Tarski system and briefly describe how it usefully generalizes the concept of semi-algebraic and, indeed, subanalytic set. No model theory will be assumed! Since there do exist excellent survey papers in this area, notably those of van den Dries, I shall keep these remarks to a minimum and my main emphasis for the rest of this lecture and the series will be on examples and methods for establishing the o-minimality of a given system of sets in real Euclidean space.

Tame Systems
Wednesday, March 12, 1997 3:30-4:30 p.m.

One of the main difficulties in establishing that a given system is o-minimal is that one has to check that a certain finiteness condition (namely the finiteness of the number of connected components) holds for sets described by arbitrary formulas of a first-order logical language. In this lecture I shall sketch a method that reduces this task to a mathematically tractable one. As a consequence one obtains what I believe to be a natural generalization of the notion of subanalytic set to the smooth context.

Approximating the Boundaries of
Closed Subsets of Euclidean Space by Smooth Manifolds

Thursday, March 13, 1997 3:30-4:30 p.m.

The title refers to the main idea involved in the proof of the result stated in the second lecture and I shall go into rather more detail here. One is looking for such an approximation technique that is inherited by both (the closure of) images under (not necessarily proper) linear maps and by intersections with affine subspaces.