Holomorphic Representations of the Braid Group
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 project will deal with certain 'holomorphic' representations of braid groups and study the dependence on an underlying 'R-matrix'.
The braid group can be intuitively understood as the collection of a finite number of strands with all possibilities of 'braiding' them, and consecutive braiding as the group operation.
This group appears in many different areas in mathematics. From its description, it is not surprising that it is relevant in knot theory. But braid groups and related algebraic concepts also appear in areas where one might not expect them, such as operator algebras, quantum field theory, or in the mathematical description of lattice systems, and have contributed to the invention of new mathematical structures such as Hopf algebras.
Project aims and methods
In applications, one is often interested in representations of braid groups, i.e. realizations of the braiding relations by linear operators on some vector space. Inspired by situations encountered in quantum field theory, a particular class of (infinite-dimensional) representations takes place on Hilbert spaces of holomorphic functions, where the entirely algebraic braid group comes into contact with concepts from (complex and functional) analysis.
Depending on the interests of the student, there are possibilities to put the emphasis abstract aspects or on applications.