C*-dynamics, quantum equilibrium, and symmetries

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The theory of operator algebras allows us to perceive and exploit a striking parallel between concrete phenomena in quantum physics, and aspects of number theory and other areas of mathematics that are very abstract, but which affect our everyday life through things like internet security and cryptography. One key common feature is the role of symmetries in physics and in number theory. Another one is that the outcome of performing successive fundamental operations depends on the order in which they are performed -this is 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. Similar parallels also exist with dynamical systems arising from stochastic processes and from fractal-like objects, in which the same shape repeats itself at smaller scales. This proposal aims to exploit these parallels by developing a framework to transfer insights and solve problems using the operator-algebraic quantum models to analyze sophisticated problems arising in number theory and dynamical systems. For instance, there is mounting evidence that the number-theoretic counterparts of physical phase transitions such as the freezing of water into ice, the spontaneous magnetization of a ferromagnet, and the motion of electrons in a two dimensional grid at very low temperature, are at the heart of the main open questions in number theory. Through the study of these operator-algebraic models, the proposed research will offer an innovative perspective into long-standing questions and will set the stage for a fascinating exchange of examples, techniques and ideas between traditionally distinct areas of mathematics. The program will expand the scope of application of operator-algebraic methods, will shed new light on open problems in mathematics, and will generate fundamental knowledge. The training of young scientists will be an important component of the proposal, which will enhance Canada's scientific capacity.*****