Tobias Fritz, Perimeter Institute for Theoretical Physics
"Real algebra, random walks, and information theory"
Similar to how commutative algebra studies rings and their ideals, the protagonists of real algebra are ordered rings. Their interplay between algebra and geometry is studied in terms of Positivstellensätze, real analogs of the Nullstellensatz, which go back to Artin's solution of Hilbert's 17th problem. I will describe some of the state of the art in this field, and then introduce a new Positivstellensatz which unifies and generalizes several of the existing ones. While traditional applications focus on polynomial rings, I will sketch a broad range of new applications in the second half, comprising random walks, asymptotic representation theory, and information theory. The first application generalizes aspects of large deviation theory to a theory of asymptotic comparison of two random walks.