Formulae of Jeffrey-Kirwan type in enumerative geometry

This thesis explores various aspects of the Jeffrey-Kirwan localisation formula, a powerful tool in computing integrals on quotients of smooth varieties by reductive group actions. Initially developed by Jeffrey and Kirwan in the symplectic context, this formula has seen various adaptations and extensions scattered throughout the literature. The primary goal of this thesis is to provide a fully algebraic proof of the Jeffrey-Kirwan localisation formula, building on the work of Lerman, Guillemin, and Kalkman. Furthermore, the thesis extends the formula to the equivariant setting, enabling the computation of equivariant integrals with respect to additional torus actions on the quotient. It also aims to clarify the relations among different versions of this formula found in the literature. In addition, the thesis explores some applications of these localisation techniques, specifically in deriving residue formulae for virtual invariants of critical loci in quotients of linear spaces, such as quiver varieties.