9314382 Staffeldt The story can be traced back to the fundamental work of Riemann, Klein, and Betti at the turn of this century, on the classification of surfaces and their higher dimensional analogues, called manifolds. Loosely peaking, they showed that for each surface there is a geometrically defined natural number called its genus, that characterizes the surface up to certain transformations, i.e. two surfaces with the same genus are indistinguishable. Moreover, given any natural number there is a model surface with that genus (to visualize a surface of genus g, imagine the surface of a coffee cup with g- 1 extra handles). A manifold of dimension n is a geometric object that looks locally like n dimensional space, in the same way that a surface looks in a small enough region, like a plane. Albert Einstein's relativity theory, with its emphasis on the four dimensional manifold of space-time, convinced the scientific world of the necessity to come to terms with manifolds of dimension greater than two. Mathematicians in the fields of algebraic and differential topology, differential geometry, among others, and physicists have devoted considerable effort in this century to developing classification schemes for manifolds of dimension three and higher. Although a great deal is known, no such schemes have been established that successfully classify manifolds of dimension three. Amazingly, in the 1960s a multistage classification scheme was developed for manifolds of dimension five and greater. The methods used to establish that scheme simply don't apply in the lower dimensions. In the early 1980s Michael Freedman proved results that led to the classification of a large natural family of four dimensional manifolds, the simply-connected ones, up to topological type, the coarsest classification that had been sought. Part of his work was devoted to constructing many manifolds not previously know to exist. Soon after Freedman produced his top ological classification theorem, Simon Donaldson used techniques of gauge theory, the subject of the conference, to shed light on a finer classification of four dimensional manifolds: the classification up to smooth equivalence, or up to diffeomorphism. Donaldson's main idea is to study the finer structure on simply connected smooth four manifolds by analyzing the geometry of spaces of auxiliary structures that can be associated with simply connected smooth manifolds. His first result was that hardly any of Freedman's new manifolds could be smooth, the main reason why it was so hard to find them. Later work has produced a host of new variants for distinguishing smooth manifolds within a given coarse, topological, Freedman type. The conference will review the properties of gauge theory which are crucial for understanding the geometry of the spaces of auxiliary structure and will clarify the connections. Then the speaker will turn to a situation which has the same formal properties of gauge theory a la Donaldson, but whose analysis is incomplete and for which several avenues for further research seem to be open.
