My task which I am trying to achieve is by the power of the
written word, to make you hear, to make you feel - it is, before
all, to make you see. That - and no more, and it is everything.
Almost two decades ago, a young mathematician by the name of Si-
mon Donaldson took the mathematical world by surprise when he discov-
ered some "pathological" phenomena concerning smooth 4-manifolds. These
pathologies were caused by certain behaviours of instantons, solutions of the
Yang-Mills equations arising in the physical theory of gauge fields.
Shortly after, he convinced all the skeptics that these phenomena rep-
resented only the tip of the iceberg. He showed that the moduli spaces
of instantons often carry nontrivial and surprising information about the
background manifold. Very rapidly, many myths were shattered.
A flurry of work soon followed, devoted to extracting more and more
information out of these moduli spaces. This is a highly nontrivial job,
requiring ideas from many branches of mathematics. Gauge theory was
born and it is here to stay.
In the fall of 1994, the physicists N. Seiberg and E. Witten introduced
to the world a new set of equations which according to physical theories had
to contain the same topological information as the Yang-Mills equations.
From an analytical point of view these new equations, now known as
the Seiberg-Witten equations, are easier to deal with than the Yang-Mills
equations. In a matter of months many of the results obtained by studying
instantons were re-proved much faster using the new theory. (To be perfectly
honest, the old theory made these new proofs possible since it created the