In brief, the student working on this project would construct skein-triangulated representations of generalised braids on homogeneous spaces of simple reductive groups.
To unpack this, the project lies on the interface of topology, geometry, representation theory, and theoretical physics. Key objects of study in low-dimensional topology are braids and links. Braids are configurations of open-ended strings with fixed endpoints, while links are similar configurations of looped strings. A crucial advance in their study was the development in 1980s of a quantum link invariant by Fields medallist Vaughn Jones. He associated a polynomial to each link via a simple procedure which repeatedly used a single skein relation to break complicated links down to simple closed loops. In late 1990s this was generalised to the celebrated Khovanov homology of links, constructed similarly using a homological version of the skein relation. In mid 2000s Khovanov homology was computed via geometric representation theory using the derived categories of certain slices of the homogeneous spaces of the group SL_n.
Logvinenko and Anno had long conjectured that skein relations could be lifted one level higher: from the level of homology to that of triangulated categories. The representations obeying these relations would be called skein-triangulated and the above should generalise to a skein-triangulated representation of generalised braids, braids with multiplicities, on the derived categories of homogenous spaces of SL_n. In 2019 this ambitious conjecture was established in its first non-trivial instance: for n=3.
The student working on this project with Logvinenko would first be trained in techniques spanning a diverse range of disciplines: algebraic geometry, representation theory, homological algebra, higher category theory, and knot theory. They would then understand and construct skein-triangulated representations of generalised braids: first for n=4 and then in full generality. They would also try to develop this theory for the simple reductive groups other than SL_n. In the process, they would interact and collaborate with the world-leading research groups in the US, Denmark, and Japan.
The 3.5 year studentship includes UK/EU fees, stipend (amount for 2020/21 is £15285) and a research training grant to cover costs such as research consumables, training, conferences and travel.
HOW TO APPLY Applicants should apply through the
Cardiff University online application portal , for a Doctor of Philosophy in Mathematics with an entry point of October 2020
In the research proposal section of your application, please specify the project title and supervisors of this project. There is no requirement to submit a research proposal.
In the funding section, please select "I will be applying for a scholarship / grant" and specify that you are applying for advertised funding from EPRSC DTP.
Shortlisted candidates will be invited to attend an interview in April
Continue with Facebook
