Werner Nahm

Many geometric problems concerning spaces of three of more dimensions have been solved by using the spaces as background for suitable physical systems. In particular one can use equations for electromagnetism or more complex gauge fields, as in the work of the Fields medalists Donaldson and Witten.

Though quantum field theory and string theory do not yet have a fully developed math- ematical foundation, they have started to be used for the same purpose. One example is the enumeration of Riemann spheres embedded in spaces of three complex dimensions by using the quantum phenomenon of mirror symmetry. The latter is based on the fact that different geometries can occur by sending the parameters of the same quantum field theory to different limits, even in cases where the geometries cannot be continuously connected in a classical way.

So far, the conformally invariant quantum field theories in two dimension have been the most useful ones. They also are the ones for which the mathematical formulation is most satisfac- tory. One of them has become famous for having the Fischer-Griess monster as symmetry group (as explored by Borcherds, who this year got the Fields medal). Further work along these lines should lead to explicit formulas for Einstein metrics on many spaces.

Traditionally, mathematicians have regarded quantum field theories as magical black boxes. To make full use of the new insights, they should be reinvented as elegant and well defined structures within the mainstream of mathematics.