Polyhedral structures on tropical varieties

Tropical varieties are polyhedral shadows of classical varieties. The purpose of these expository notes is to explain the origin of this polyhedral complex structure from the perspective of Gr\"obner bases. To appear in the proceedings of the 2011 Bellairs Workshop in Number Theory

Lower and upper bounds for nef cones

The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space \bar{M}_{0,n} to a wide class of varieties.Comment: 25 pages, 4 figures. Final version to appear in IMR

Higher Laminations and Affine Buildings

We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and $PGL_m$. The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of laminations. Our construction involves positive configurations of points in the affine building. We show that these laminations are parameterized by the tropical points of the spaces \X_{G,S} and \A_{G,S} of Fock and Goncharov. Finally, we explain how these laminations give a compactification of higher Teichmuller spaces.Comment: 46 page