Quantum Mathematics and Geometry

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The proposed research will explore the mathematical ideas of quantum computation and quantum information theory, and related ideas in mathematics itself. A defining feature of the mathematics of quantum theory is non-commutativity, the idea the "A times B" does not always equal "B times A". This is reflected in experiments in quantum mechanics that the outcomes of measurements depends on the order in which they are measured; moreover, measurements in quantum mechanics are inherently statistical. The phenomenon of non-commutativity leads to surprising new algorithms that could run on quantum computers if they are one day constructed. In mathematics itself, quantum-style non-commutativity has led to many new ideas in geometry, such as new notions of symmetry, and in topology, such as new methods to describe and distinguish knots and similar topological objects. The PI plans to investigate connections and new constructions among all of these types of quantum mathematics.