Operator algebras and noncommutative geometry
Operator algebras and noncommutative geometry has developed in the last thirty years from a branch of functional analysis into a central field in mathematics.
It has application and connections with algebra, analysis, geometry, typology and statisitcal mechanics and conformal quantum field theory in mathematical and theoretical physics.
In 1982, Vaughan Jones introduced his subfactor theory. A factor is an infinite dimensional algebra of operators closed under natural algebraic and topological operations and having no non-trivial ideals – analogous to a simple group.
An outer group action on a factor can be recovered from the inclusion of the fixed-point algebra in the original factor. A subfactor or an inclusion of one factor in another is then a generalisation of a group or a quantum symmetry. A subfactor encodes the symmetry. Dimension is ubiquitous in mathematics. The Jones index measures the relative dimension of the larger algebra over the smaller. K-theory also provides tools for understanding dimension and both of these notions are at the heart of this project.
One tantalising problem is to understand if there is anything beyond classical groups and their deformations as symmetries on an algebra of operators
The project will involve one or more aspects of the following: structure theory of operator algebras, C*-algebras, von Neumann algebras, factors, subfactors, planar algebras; quantum groups and Hopf algebras; non-unitary theories, Leavitt algebras, logarithmic and non-semisimple systems; twisted equivariant K-theory and bivariant Kasparov theory; higher twists and higher geometry.
We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.
Please contact the supervisor when you want to pursue this project, citing the project title in your email, or find out more about our PhD programme in Mathematics.