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Monday, April 24, 2017

**Abstract:** Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer $\mathcal X$ is the family of centralizers in $G$ of regular elements in Lie($G$). This algebraic variety has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^ ∗G$. We introduce a relative compactification of $\mathcal X$ , in which every centralizer fiber is replaced by its closure in the wonderful compactification of $G$. We show that the symplectic structure extends to a log-symplectic structure on the boundary, using the logarithmic cotangent bundle of the wonderful compactification.