Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties

We show, using a direct variational approach, that the second boundary value problem for the Monge-Amp\`ere equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that (X,D) admits a (singular) K\"ahler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Zhou-Wang concerning the case when X is smooth and D is trivial. Li's toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain K\"ahler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the K\"ahler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu's boundary value problem on the convex body P. in the case of a given canonical measure on the boundary of P.Comment: 53 page

Las Vegas and Houston: Global Command Centers in the Sun Belt

Apart from their notoriously hot summers, Las Vegas, Nevada and Houston, Texas appear to have little in common. Upon further scrutiny, however, the two cities have followed strikingly parallel trajectories. As a hub of commercial gaming today, Las Vegas faces challenges comparable to the obstacles Houston encountered in the 1970s and 1980s as a hub of oil and energy. The story of Houston since this downtime reflects a stunning transformation into a modern-day international city, and one that both parallels and portends a new Las Vegas. In this article, we argue that Houston provides a viable model for Las Vegas as a “global command center” of a major international industry – and that in many ways Las Vegas is already following this path