Abstract : I will report recent progress on the cohomology of the moduli of one-dimensional sheaves on
the projective plane. This moduli space relates intimately to the enumerative geometry of
local P^2 via a certain perverse filtration on its cohomology, which is in turn determined by
the cohomology ring structure. I will explain a systematic approach to study this cohomology
ring. Our main ingredients include Mumford-type geometric relations and a representation of
the (half) Virasoro algebra on the cohomology of the moduli spaces. Time permitting, I will
also discuss a conjectural description of the perverse filtration in terms of explicit tautological
classes. Based on joint work with Y. Kononov, W. Lim, and M. Moreira.