The theory of operator algebras originated with von Neumann in the early part of the 20th century as a mathematical framework for modeling phenomenon in quantum mechanics, and has since become a fundamental component of modern theoretical physics, most recently as the primary setting for quantum information theory. In addition, beginning with the surprising subsequent discovery of continuous dimensionality, a feature which radically distinguishes their geometry, the theory quickly started to reveal its rich potential as an abstract framework for exploring new kinds of symmetry and structure that connect it to several other areas of mathematics, including ergodic theory, harmonic analysis, logic, and group representation theory. Over the last eighty years this potential has been borne out in spectacular and often unexpected ways, and the connections have broadened even further to encompass knot theory, algebraic and differential topology, geometric group theory, quantum groups, conformal field theory, number theory, probability, and quantum information.

The field of operator algebras has ballooned and diversified, so that there is enormous merit to an extended program which comprehensively brings together the strands. While a three-week meeting at Queen's University, Kingston in 1980 was ample enough at the time to allow for all of the major developments in operator algebras and their applications to be presented and discussed, nowadays the field is so expansive that most of the major research events are either somewhat specialized (for example, to one of the major branches|C*-algebras or von Neumann algebras) or permit only a big-picture presentation of key advances across sub fields. At the same time, the various streams can benefit from an interchange of ideas and techniques guided by an allembracing vision of the subject. 

At the same time the various streams can benefit from an interchange of ideas and techniques guided by an all-embracing vision of the subject. The proposed six-month program will serve to strengthen the links both within operator algebras and from operator algebras outwards with emphasis on the following themes, given in expected chronological order:

• Connections with Logic
• Symmetry and Structure
• Groups and Group Actions
• Free Probability
• Non-Commutative Geometry

Research in groups for underrepresented genders in operator algebras

Funding is available for small groups of operator algebraists that are underrepresented or in minority within mathematics due to gender (including, but not limited to, cis woman, transgender, and non-binary) to work together at the Fields Institute at a mutually convenient time. Groups interested in this opportunity should contact the thematic program organizers.