The wonderful compactification and the universal centralizer

This thesis describes work of the author involving the geometry of the wonderful compactification of a complex semisimple algebraic group G of adjoint type. We consider the centralizer Ge in G of a regular nilpotent element e ∈ Lie(G), and we prove that its closure in the wonderful compactification is isomorphic to the Peterson variety. We generalize this result to show that the closure in the wonderful compactification of the centralizer G^x of any regular element x ∈ Lie(G) is isomorphic to the closure of a general G^x-orbit in the flag variety.,We then consider the family X of all closures of such centralizers, parametrized by conjugacy classes of regular elements in Lie(G). This is a relative compactification of the universal centralizer X , which has a natural symplectic structure arising from a Hamiltonian reduction of the cotangent bundle of G. We prove that the symplectic structure on X extends to a log-symplectic Poisson structure on X, by realizing X as a Hamiltonian reduction of the logarithmic cotangent bundle of the wonderful compactification.