Holomorphic Representations of the Braid Group
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 project is advertised as part of the EPSRC Doctoral Training Partnership. It is currently not available to self-funded applicants.
Closely related to the symmetric group, the braid group appears in many different areas in mathematics. From its description, it is not surprising that it is relevant in knot theory, where braids are linked to the famous Jones polynomial. But braid groups and related algebraic concepts also appear in areas where one might not expect them, such as operator algebras, quantum field theory, abstract categories, or in the mathematical description of lattice systems (such as magnets), and have contributed to the invention of new mathematical structures such as Hopf algebras.
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.
In general, the representation theory of the braid group is not yet well understood. The project will deal with a particular case of "holomorphic" representations and study the dependence of such representations on an underlying "R-matrix" or "Yang-Baxter operator".
Depending on the interests of the student, there are possibilities to put the emphasis either on purely mathematical aspects, or also to include applications.