Operator Modules, Quantum Groups and Quantum Information

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In 1925 our view of the physical world drastically changed with the advent of Heisenberg's matrix mechanics. He showed that we may accurately describe quantum phenomena by interpreting time dependent variables as non-commuting matrices rather than functions. This "quantization" of functions, which underlies the theoretical development of quantum mechanics, has motivated mathematicians to quantize other areas of mathematics, including functional analysis, harmonic analysis, and information theory. The resulting areas: operator spaces, topological quantum groups, and quantum information theory, are, to this day, prominent world-wide research areas at the forefront of modern analysis and mathematical physics. They have all been shown to have a profound structure theory, and deep mathematical connections between them continue to emerge. My research program lies at the confluence of these three areas.An area where the above fields interact in a fruitful manner is the current rapid development of harmonic analysis on quantum groups. The operator spaces of interest in this theory carry a natural module structure over a non-commutative algebra, and, in fact, the corresponding operator module structure is fundamental to the theory. Analogous to the quantization of functional analysis to the analytical theory of operator spaces, the main long-term vision of my research program is the development of the analytical theory of operator modules and their applications to non-commutative harmonic analysis. This development will create an entirely new facet of modern functional analysis with promising applications to quantum group theory, including the potential resolution of several important open problems. The original techniques outlined in the proposal have already had a significant impact on quantum group theory, and their development will continue to furnish the theory with novel tools for its evolution.Another long-term goal of the proposed research is the development of new applications of operator spaces and harmonic analysis to quantum information. These areas have already provided valuable tools for quantum information theory, and it is of great interest to explore further connections between them. In particular, we aim at exploring a recent connection between non-commutative harmonic analysis and the fundamental structure of quantum entanglement, which is intimately related to one of the biggest open problems in operator algebras.The interdisciplinary research in this proposal, together with my interdisciplinary background, provide excellent training opportunities for HQP at all levels. Such an interdisciplinary approach fosters the rapid mathematical maturity of HQP as well as the development of versatile skills, techniques and perspectives, which provide them with exceptional preparation for future endeavors within several interacting research areas.