Tilings and mutation

Supervisor(s)

Contact Professor Karin Baur to discuss this project further informally.

Project description

Mutation is the key notion in cluster theory, most prominent is the exchange notion arising from flips in triangulations (Fomin-Zelevinsky 2002, Fomin-Shapiro-Thurston 2008). Relying on Hatcher's result (Hatcher1991) that for any pair of triangulations of a given surface, there exists a sequence of flips connecting them. Triangulations of surfaces are essential in defining cluster categories. Generalizing this, quadrangular tilings of surfaces are models for higher cluster categories, with rotation replacing flips.

The interplay between surface geometry and algebraic questions is further illustrated by the use of decorated tilings in the context of bases for (coloured) Temperley-Lieb algebras, cf. Marsh-Martin 2010, DiFrancesco1998. The decorations I have in mind are pairs of non-crossing curves, linking midpoints of neighboured edges.

The student will start from here and use a new kind of decorated tilings. The idea is to work with a pair of oriented three-legged trees on rhombic tilings, connecting mid-points of three successive edges of the tile. I expect that this will result in Kuperberg spiders, Kuperberg 1996, which are essential in the representation theory of $U_q(\mathfrak{sl}_3)$. The student will explore this with the goal to provide bases of Kuperberg spiders.

Entry requirements

Applicants should have, or expect to obtain, a UK first class honours degree in Mathematics or a related discipline, or equivalent.

If English is not your first language, you must provide evidence that you meet the University’s minimum English Language requirements.

How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is 'Tilings and mutation’ as well as Professor Karin Baur as your proposed supervisor.

We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates.  All scholarships will be awarded on the basis of merit.

If you require any further information please contact the Graduate School Office, e: maps.pgr.admissions@leeds.ac.uk.