Quantum Methods in Mathematics and Computation

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The aim of this project is to explore quantum methods, or methods of non-commutative algebra and non-commutative probability, in traditionally non-quantum settings. In particular, the project will explore special bases of invariant tensors, hyperbolic structure as a limit of quantum 3-manifold invariants, quantum algorithms, and notions of quantum metric spaces. Related spin-off projects include an extension of the probabilistic method in probability using the central limit theorem, and a classification of known relations among complexity classes (both classical and quantum).The aim of this project is to apply the ideas of quantum physics to both mathematics, in particular to geometry and topology, and to computer science. The heart of quantum mechanics is the statistical notion of quantum uncertainty. This is the idea and the fact that when one physical measurement of an object (such as position) is known precisely, other related values (such as momentum) cannot have definite values, but instead have a statistical range of values. Quantum uncertainty is always modelled by the mathematical notion of non-commutativity, which is the idea that AB need not equal BA when A and B are abstract variables. Non-commutativity plays an increasingly important role in low-dimensional geometry and topology. Quantum uncertainty could also lead to new kinds of computers that can execute special quantum algorithms. These two currents of research are intellectually similar, and this project will explore them together.