However, in the last two years we have witnessed dramatic developments bringing for the first time nonperturbative questions into reach. In a confluence of a wide variety of ideas, many of them dating back to the 70s and 80s, the structure, internal consistency and beauty of string theory has greatly improved. We have now a much better and clearer picture what the theory is about: it is well-defined and unique!
Crucial in all this has been the concept of string duality. In
fact, duality has been a powerful idea in physics and mathematics for a
long time. A very elementary illustration can be given by considering
simple linear duality of a vector space, or even better, of a lattice
. The theta-function
satisfies by Poisson's formula the relation 
with 
the dual lattice. So the
expansion of 
for small values of 
, can be
directly translated into an expansion of the dual function
for large values of and vice versa.
Physically, the transformation to a dual set of variables can translate
a difficult question (such as strong-coupling behaviour with strong
quantum fluctuations for large ) into a much more accessible one
(weak-coupling behaviour where the semi-classical approximation for
small makes sense). String duality is the statement that the
above string theory functors 
have precisely the same
properties as the simple theta-function we discussed above! Furthermore,
all five diffferent perturbative strings are related in such a fashion
and are just expansions of one single unified theory around different
backgrounds.
The mathematical applications of string duality are just in their
infancy. For example, one of the consequences is the realisation that
string theory does not only include strings but also various higher
dimensional objects, known as D-branes. A D-brane is simply a place in
space-time where the string can begin or end. So, instead of considering
maps of a Riemann surface 
into a space-time manifold M, we
consider a submanifold 
(the D-brane) and maps 
with the property that the boundary of the surface lies
on the brane, as seen in figure 2. These cycles satisfy special conditions on
the embedding and define a moduli space (typically a moduli space of
sheaves). The D-brane quantum states should then roughly correspond to
the homology of this moduli space. String duality can now be used to
argue that in certain cases the Hilbert spaces of D-branes are
isomorphic to the Hilbert spaces 
of strings that we introduced
earlier. Thus two very different subjects are related: moduli spaces of
vector bundles or sheaves and infinite-dimensional algebras! These
connections indicate beautiful new mathematical structures and are now
actively investigated.