Vladimir Dotsenko (Dublin Institute for Advanced Studies)
We introduce several associative algebras and series of graded vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we construct a series of representations of symmetric groups which turn out to be isomorphic to the so called parking function modules. We also construct --- in a similar fashion --- series of graded vector spaces whose dimensions are Catalan numbers and Fuss--Catalan numbers respectively. Conjecturally, these spaces deform the spaces of global sections of certain vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.