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1. Limit Sets of Weil–Petersson Geodesics

   https://doi.org/10.1093/imrn/rny002  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (February 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we prove that the limit set of any Weil–Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil–Petersson geodesics with minimal non-uniquely ergodic ending laminations and limit set a circle in the Thurston compactification. 

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2. The partial compactification of the universal centralizer

   https://doi.org/10.1007/s00029-023-00873-8  

   Bălibanu, Ana (November 2023, 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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3. Steinberg slices and group-valued moment maps

   https://doi.org/10.1016/j.aim.2022.108344  

   Bălibanu, Ana (June 2022, 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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4. The birational geometry of moduli of cubic surfaces and cubic surfaces with a line

   https://doi.org/10.1112/mod.2024.12  

   Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus (January 2025, Moduli)

   Abstract We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubic surfaces. 

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5. Limit sets of Weil–Petersson geodesics with nonminimal ending laminations

   https://doi.org/10.1142/S1793525319500456  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (March 2020, Journal of Topology and Analysis)

   null (Ed.)

   In this paper, we construct examples of Weil–Petersson geodesics with nonminimal ending laminations which have [Formula: see text]-dimensional limit sets in the Thurston compactification of Teichmüller space. 

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