Lie groups and twisted K-theory
This research project is in competition for funding with one or more projects available across the EPSRC Doctoral Training Partnership (DTP). Usually the projects which receive the best applicants will be awarded the funding. Find out more information about the DTP and how to apply.
Application deadline: 15 March 2019
Start date: 1 October 2019
The main goal of algebraic topology is to associate algebraic structures to topological spaces that remain unchanged if the spaces are deformed continuously. Understanding these topological invariants often gives rise to powerful classification results with applications in other research areas like algebra or mathematical physics.
Generalised cohomology theories are particularly well computable invariants and topological K-theory belongs to this class. It is a cohomology theory built from the algebra of vector bundles over a space.
A variant of this theory, called twisted K-theory, relies on extra structure - called a twist - usually encoded in a bundle of algebras or a gerbe over the space. It should be thought of as the K-theoretic analogue of ordinary cohomology with local coefficients. Equivariant twists over Lie groups gained increasing importance due to a deep theorem by Freed, Hopkins and Teleman, which relates the twisted equivariant K-group of a Lie group to the Verlinde ring of positive energy representations of its free loop group. In case of the groups SU(n), there is a natural generalisation of the twist considered by these authors, which involves exponential functors on finite dimensional complex vector spaces.
Project aims and methods
This project deals with the computation of the characteristic classes of these twists in rational cohomology. Partial results in this direction have already been achieved in a preprint. The generators of the cohomology rings of the (special) unitary groups can be detected via natural maps from the suspension of complex projective space to the groups. The project involves studying generalisations of those maps from projective spaces to Grassmann manifolds. Continuous maps of this type appeared in several other places in algebraic topology as well, most notably in the work of Miller on the stable splitting of the unitary groups. The successful applicant will have the opportunity to explore those connections.