Virtual classes and enumerative invariants

Enumerative geometry is a branch of classical algebraic geometry whose
aim is to count the solutions to a given problem. As an example, given
four general lines in 3-space, there are exactly two other lines
meeting each of them simultaneously. The solution to such a problem is
usually found by constructing a "moduli space" (in the example the
Grassmann 4-manifold G of lines in 3-space) and then doing
"intersection theory" on it (computing the number of points, or
degree, of the intersection of M(L_I). I=1,...4, where M(L) is the cod
1 submanifold of G parametrizing lines that meet a given line L).

In general, e.g.to address problems coming from gauge and string
theory, one needs (a combination of) three kinds of generalizations:
1) If the moduli space is not compact, a geometrically meaningful
compactification must be found;
2) If the moduli space is singular, i.e. not a manifold, something
must be done to make intersection theory possible;
3) If the moduli space doesn't behave well more general objects
(moduli stacks) are needed.

We will give examples of situations 1) and 3), and concentrate on
problem 2); we will explain what is a virtual class, how one can find
one and use it to define invariants and (sometimes) compute them. We
will also sketch how it relates to other approaches to solving problem 2).