Operator Algebras and Mathematical Physics

Date: 29 September - 3 October 2025

Location: ICMS Edinburgh

Quantum Field Theory (QFT) stands as one of the pivotal theories in physics, holding a significant position over a multitude of subjects. Within various mathematical approaches to QFT, we emphasize the operator algebraic methods. Ever since von Neumann’s initiation, the theory of operator algebras has been closely related to quantum physics. Jones’ work on subfactors, including its application to low-dimensional topology such as the Jones polynomial, has produced a substantial influence across a broad spectrum of mathematical and physical topics.

Two-dimensional conformal field theory serves as a particularly fertile example of a QFT. It represents one of the rare instances where everything can be treated with mathematical rigor and the operator algebraic approach has been quite successful.

Von Neumann algebras have also gained interest recently in the context of entanglement entropy calculations. Bringing quantum reference frames and gravity into the description of quantum fields affects what physical observables one considers. This is tightly connected to von Neumann algebra types and it is timely to provide a link between the existing physical examples and rigorous results in mathematics.

Both classical and quantum statistical mechanics have also had interactions with operator algebras over decades. Profound connections to subfactor theory are exemplified by the Temperley-Lieb algebra since early days and recent advances in topological phases of matter, a hot topic in contemporary physics, have led to even deeper relations with operator algebras.

With the rapid advancements in operator algebras and related areas in mathematical physics, understanding the complete landscape is a challenge for both senior  and junior researchers. This presents a perfect moment for this workshop, where experts from various fields can converge and acquaint younger researchers with these dynamic domains.

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