Cartan Subalgebras in Operator Algebras, and Topological Full Groups (24w5175)


Astrid an Huef (Victoria University of Wellington)

(University of Victoria)

(University of Wollongong)

Dilian Yang (University of Windsor)


The Banff International Research Station will host the “Cartan Subalgebras in Operator Algebras, and Topological Full Groups” workshop in Banff from November 3 - 8, 2024.

The mathematics known as ``operator algebras'' was invented as a tool for understanding the complexities of quantum mechanics. Today it underpins the mathematical framework for quantum statistical mechanics, much of what we know of so-called topological states of matter, and much of quantum computing. These operator algebras are, in general, extremely complicated and abstract, so in order to understand them and to come up with examples that provide good models for quantum systems of interest, we need a way to visualize and work with them, much like sketching the graphs of functions helps us to visualize and understand the complexities of calculus. A number of recent significant breakthroughs have demonstrated that structures called "Cartan subalgebras'' and "groupoids'' can provide these much-needed visual models for operator algebras. Meanwhile, the mathematics of "group theory'' underlies our understanding of symmetry and structure in both mathematics and the world around us, and is deeply entwined with concepts ranging from our understanding of the periodic table of elements all the way through to conserved quantities in quantum systems. Groups have been used in mathematics as far back as the mid 19th century, but they still present many mysteries.

This workshop is inspired by several recent breakthroughs in the mathematics of Cartan subalgebras and groupoids. These results showed that understanding Cartan subalgebras could unlock the last remaining mystery at the heart of classification of $C^*$-algebras, could answer long-standing questions in the study of complex systems evolving through time, and have already provided new examples in group theory that have evaded researchers for many years. The workshop will provide a forum for interaction between these research areas to capitalize on the unexpected new connections arising from these new results. To do so, it will bring together the best minds, from early-career researchers to current world-leaders in the three areas in question. It will address major open problems, and will forge new collaborations that will carry mathematical research forward over the coming years.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta’s Advanced Education and Technology, and Mexico’s Consejo Nacional de Ciencia y Tecnología (CONACYT).