SDU Operator Algebras Seminar

Title: Quantum graphs and spin models

Abstract: Spin models for singly-generated Yang-Baxter planar algebras are known to be determined by certain highly-regular classical graphs such as the pentagon or the Higman-Sims graph, which are extremely rare. Examples of spin models include the Jones and Kauffman polynomials. We will discuss the notion of higher-regularity for quantum graphs and give new examples of non-classical graphs yielding spin models. Time allowing, we will discuss some applications to quantum groups, topology and quantum information. 

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Title: Selfless Inclusions of C*-Algebras and Quantum Groups

Abstract: Recently, strong asymptotic freeness, or selflessness, in C*-algebras has emerged as a powerful technique to prove important regularity properties including simplicity, unique trace, stable rank 1, and strict comparison. In particular, in Fall 2024, Amrutam, Gao, Kunnwalkam Elayavalli, and I showed that the reduced group C*-algebras of all hyperbolic groups with trivial finite radical are selfless, which resolved the open problem of strict comparison for the reduced group C*-algebra of the free group on two generators. Since then, our result has been expanded to include a much larger class of groups. Work has also begun on isolating selflessness for C*-algebras not arising from groups, including the result of Hayes, Kunnawalkam Elayavalli, and Robert on selflessness of the reduced free product of a large class of C*-algebras (see also Flores-Klisse-Ó Cobhthaigh-Pagliero). I will introduce the general notion of a selfless inclusion of C*-algebras, with which we will see the selflessness of the reduced unitary compact matrix quantum groups. This work is joint with Ben Hayes, Srivatsav Kunnawalkam Elayavalli, and Leonel Robert.

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