Wednesday, November 18, 2020 at 12:00pm to 1:00pm
Aaron Lauda, USC
Abstract: Quantum groups describe a rich class of symmetries that manifest in a variety of mathematical and physical situations. This talk will provide intuition for quantum groups and survey how these symmetries appear in exactly solvable lattice models, 3-dimensional topological quantum field theory, and fault-tolerant approaches to quantum computation. Intertwined with all these constructions is a rich class of knot invariants, including the famous Jones polynomial. We will explain how much of this web of interconnected ideas is just the shadow of a far richer theory involving more sophisticated notions of symmetry and more advanced physical theories. We will introduce the concept of 'categorification' and illustrate how this perspective reveals a new layer living above these constructions. At the heart of the categorification philosophy is the idea that by accessing this higher level structure living above an existing theoretical framework, one can lift the scope of applicability for the theory, lifting knot invariants to more powerful invariants, 3D TQFTs to 4-dimensional theories, and 2D lattice models to 3D. If time permits, we will explain how the 4-dimensional topological theories allow for new possibilities for topological quantum computation.
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