Homological study of some Hilbert schemes
This lecture gives a brief historical survey of the Hilbert schemes of points on a surface through cohomological considerations. If the first steps in the theory of Hilbert schemes are based on techniques of pure algebraic geometry, the representation theory is widely used thereafter to understand the cohomology groups : this is the idea of Nakajima, who uses representations of Heisenberg-Clifford algebras.
We can then seek to generalize this method : to an object from algebraic geometry, we try to give a structure of module over a certain type of algebra whose representations can be studied separately. In this sense, Schiffmann and Vasserot studied the K-theory of the Hilbert scheme of points on the plane using elliptic Hall algebras and Hecke algebras.