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  1. (, Selecta Mathematica)
    The universal centralizer of a semisimple algebraic group is the family of centralizers of regular elements, parametrized by their conjugacy classes. When the group is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification of the universal centralizer by taking the closure of each fiber in the wonderful compactification. We use the geometry of the wonderful compactification to give an explicit description of the symplectic leaves of this new space. We also show that its compactified centralizer fibers are isomorphic to certain Hessenberg varieties—we apply this connection to compute the singular cohomology of the compactification, and to study the geometry of the corresponding universal Hessenberg family. 
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  2. ; (, Transformation Groups)
    We use the Springer correspondence to give a partial characterization of the irre- ducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kahler package.” 
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  3. (, Advances in Mathematics)
    We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification of this space by taking the closure of each centralizer fiber in the wonderful compactification. By realizing this relative compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic. 
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