The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as Mg ,w for a particular genus g and a wei...

We investigate uniqueness issues that arise in l∞-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of R based on the dimension of the set of l∞-nearest neighbors. This implies that the l∞-nearest neighbor in a linear space is unique if and only if the underlying matroid is uniform. For Bergman fans of matroids, we show that the se...

Let L be a holomorphic line bundle on a compact complex manifold X of dimension n, and let e be a continuous metric on L. Fixing a measure dμ on X gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of L. We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair (K,φ), ...

Our main goal in this paper is to construct the first explicit fundamental domain of the Picard modular group acting on the complex hyperbolic space CH . The complex hyperbolic space is a Hermitian symmetric space, its bounded realization is the unit ball in C equipped with the Bergman metric. The Picard modular group is a discontinuous holomorphic automorphism subgroup of SU(2, 1) with Gaussia...