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Atranslationsurfaceisamultifacetedobjectthatcanbestudiedwiththetoolsofdynam- ics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces.
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We exhibit an infinite family of discrete subgroups of Sp4 (R) which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations. Additionally, we relate the cones used for ping-pong in R4 with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1).more » « less
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We construct canonical positive currents and heights on the bound- ary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous fam- ily, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.more » « less
