"The interplay between analysis and topology in some nonlinear PDEs"

We will discuss several modes of interaction between analysis and topology which have led to fruitful developments in recent years. Singularities are observed in a number of physical problems, e.g. liquid crystals, superconductors, superfluids,etc....; such singularities can be isolated points, or distributed along lines or surfaces. Their modeling is described in terms of nonlinear PDEs(stationary or evolution) where the unknown is a map taking its values into a manifold- for example a sphere.

The occurrence and classification of singularities inside a domain can be predicted from the topological considerations. In some cases, the creation and distribution of singularities inside a domain can be predicted from the topological properties of the (smooth) boundary data.in other cases, point - -singularities in the boundary condition "propagate" inside the domain in the form of line-singularities. The Sobolev spaces$W^{1,p}(M,N)$,where $M$ and $N$ are manifolds, play a central role in the formulation of such problem.

Natural topological questions concerning these spaces have been tackled only recently and many problems are still open. For example, when $p\geq$ dim $M$ the homotopy classes are the same as the standard ones. However when $p$ decreases the topology "deteriorates", and for $p,2$ any two maps are homotopic in $W^{1,p}$. In the intermediate range $2\leq p,$ dim $M$ some homotopy classes may disappear while others persist.