Quantum equilibrium and symmetries from number theory

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The proposal seeks to develop a recently established connection between operator algebras, originally developed to model concrete quantum mechanical phenomena, and abstract algebraic structures arising in number theory. This remarkable connection is based on the fact that numbers and sub-atomic particles share some common features that make them tractable with the same mathematical tools. One of these common features is the prominent role that symmetries play in their study; another one is the relevance of operations that do not commute with each other --the underlying phenomenon behind Heisenberg's uncertainty principle. In quantum systems the non-commuting operations are the measurements of position and momentum of particles; in numerical systems, they are the two basic operations of addition and multiplication. This line of thought reveals that concrete phenomena such as the freezing of water into ice, or the spontaneous magnetization of a ferromagnet, have surprisingly sophisticated abstract analogues. It turns out that when we let number systems 'freeze' and then analyze the equilibrium and the symmetries that are spontaneously broken in the process, or when we let them 'boil' and then analyze the ensuing chaotic behavior, we are led to key questions in number theory viewed from a very innovative perspective. This sets up a fascinating exchange of examples, techniques and insight between traditionally distinct areas of mathematics. The program falls within the general theory of operator algebras, in which Canada is an international leader. The expected outcomes will broaden the scope of their application, will shed light on long-standing problems, and will generate fundamental knowledge, thus enhancing Canada's presence in the international scientific arena.