Over the last decade, convexity has re-emerged as an powerful tool linking algebraic geometry and combinatorics. Although the connections between polyhedral and toric geometry are very well-known, the recent results exploiting convexity recognize new combinatorial structures on a much larger class of algebraic varieties. This work is especially striking for real varieties. In the other direction, spectacular progress in combinatorics has just been achieved by importing convex machinery from algebraic geometry. The primary scientific aim of this workshop is to explore these significant developments.

The focus of the meeting will be on identifying emerging trends, discussing work in progress, and above all, starting new projects and collaborations. We expect this conference to influence and initiate the next advances at the interface between algebraic geometry, combinatorics, and convexity.