A famous theorem of Mather shows that the topologically stable maps lie dense in the space of all smooth maps with compact source. Sadly, the theorem is non-explicit - it is extremely hard to identify topologically stable maps on the basis of Mather's work.

More recent work of myself and C. T. C. Wall has removed much of the difficulty, by comparing the geometric and algebraic properties of map-germs which are necessary for topological stability to those which can be proved to be topologically stable by some new analytic techniques. The result is a computable characterisation of topological stability in a very wide range of dimensions.

The lectures will be an introduction to the ideas involved in these results.