Given a complex algebraic surface X, the associated Hilbert scheme is a collection of projective varieties which parameterizes the subvarieties of X.

Recently, Hilbert schemes have made a rather surprising appearance in a variety of different contexts.

To simplify a little, the rational cohomology groups of the Hilbert scheme are naturally the standard representation of a certain infinite dimensional (super) Lie algebra, called the Heisenberg/Clifford algebra. In the hyperkähler case, which is the one of interest to physicists, the character formulas of these representations are modular forms.

This talk is devoted to explain the picture outlined above and to describe my contribution to the subject.

The cohomology groups mentioned above, though transcendental in nature have been computed using the purely algebraic method of reduction to finite fields (Weil Conjectures).

My contribution is a new, simpler and geometric method, suggested by the presence in the picture of the Heisenberg algebra, to calculate these cohomology groups. In particular, the method works for not necessarily algebraic surfaces.

Applications and open questions will be discussed.