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Explore the mathematical foundations of planar algebras in this 58-minute seminar lecture from the BIMSA-Tsinghua Quantum Symmetry Seminar. Delve into the study of generalized symmetries of subfactors and their encoding through planar algebras, examining the fundamental question of what minimal structure can be universally expected from these symmetries. Learn how Jones demonstrated that planar algebras associated with arbitrary subfactors contain the Temperley-Lieb-Jones algebra, and discover how the presence of intermediate subfactors leads to the inclusion of the Fuss-Catalan-Bisch-Jones algebra. Focus on the special case of cocommuting squares of factors with group-like properties, where the speaker presents novel research on skein relations of their associated planar algebras and demonstrates how these algebras extend partition algebras. Gain insights into this collaborative research with Dietmar Bisch that addresses the open problem of subfactors with two intermediate subfactors, contributing to the broader understanding of quantum symmetries in operator algebra theory.
