9626562 Pedersen This project is mainly concerned with manifolds and their symmetries. A manifold is a second countable Hausdorff topological space which is locally homeomorphic to euclidean space. There is quite a difference in studying manifolds in high dimensions and in low dimensions, and this project is mainly concerned with high dimensional phenomena. The non-linear similarity problem is the question whether two representations of finite groups can be homeomorphic without being linearly ismorphic. This question is high-dimensional in nature, since it is easy to see that such phenomena cannot occur in dimensions lower than four. The other main part of this project is the Novikov and Borel conjectures. These conjectures are concerned with classification of manifolds in high dimensions. Classification is, of course, interesting in all dimensions, but the methods are quite different in high dimensions and in low dimensions. The project has connections to many branches of mathematics, notably to analysis. The main methods used are from controlled topology, and there are analogues of controlled topological problems in analysis, notably through the Baum-Connes conjectures. While high dimensional manifolds are not readily accessible to our mind's eye, they arise naturally in real-life situations whenever there are numerous degrees of freedom, as, for example, the space of all possible configurations of a mechanical linkage with numerous joints and pivots. ***
