Tobias Fritz, Perimeter Institute
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 Positivstellen- stze, 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 eld, and then introduce a new Positivstellensatz which unies 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 rst application generalizes aspects of large deviation theory to a theory of asymptotic comparison of two random walks. applied e.g. to reprove the known classification of pure bipartite entanglement in the asymptotic or catalytic setting. I will also present the resource theory of quantum thermodynamics without infinite bath. It allows for the analysis of heat engines with finite (but large) reservoir in terms of simple planar geometry.