The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz.  Over the last two decades the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged.  In this talk I'll discuss commuting operator strategies for nonlocal games, recall NC Nullstellensatz which are helpful, and then apply them to a very broad collection of nonlocal games.

The main results of this procedure will be two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. The second characterization is based on a new Nullstellensatz. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem. Time permitting, I'll also discuss how to recover some standard characterizations of perfect commuting operator strategies, such as the synchronous and linear systems games characterizations, from the Nullstellensatz formalism.

This talk is based on joint work with John William Helton and Igor Klep. arXiv link to appear soon.