*Thematic Program on Singularity Theory and
Geometry *

January - June 1997

## Coxeter Lecture Series

### Alex J. WILKIE,

Mathematics Institute, University of Oxford

### A series of three lectures on

O-Minimal Tarski Systems: Theory and Examples

**Introduction**

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.