Geometry of moduli spaces and non-abelian localization formal

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We have been studying the topology of hyperKahler quotients and symplectic quotients. The aim of the research is to understand the structures of various important moduli spaces, because many of these spaces are constructed as such kind of quotients. Recently the topology of symplectic quotients has been studied intensively by using Morse theory and equivariant cohomology theory. However, since hyperKahler quotients are non-compact, these methods do not work in general. So little is known about the topology of hyperKahler quotients.In the first year of this project we investigated many examples of hyperKahler quotients by tori. In the second year we proposed a conjecture about the ring structure of their cohomology and gave a partial answer. In the last year we proved the conjecture affirmatively. A hyperKahler quotient contains a union of symplectic quotients as its deformation retract. To prove the conjecture, it is important to show that these symplectic quotients intersect in a simple way.We also investigated some examples of hyperKahler quotients by non-abelian groups, especially a partial compactification of the cotangent bundle of the configuration space of points in the projective line. As a result we calculated the generating function of the intersection parings on the configuration spaces.