Project 12: "Exchange graphs of triangulations"

This project is running since 2015.

Principal investigator: Karin Baur
University of Graz, Austria.
Mentor for: Lindorfer, Oh.

DK Student

  • Second phase of the doctoral program:
  • Lukas Andritsch (Austria; since October 2015)
    Mentors: Oswin Aichholzer, Daniel Smertnig.

Associated Students

  • Second phase of the doctoral program:
  • Jordan McMahon (New Zealand; since October 2015)
    Mentors: Alfred Geroldinger, Dusko Bogdanic.

  • Hannah Vogel (USA; February 2013 - September 2016)
    Mentors: Michael Kerber, Birgit Vogtenhuber.
  • PhD Defense: September 08, 2016.
    Referees: K. Baur, A. Felikson (Durham)
    Examiners: K. Baur, A. Felikson (Durham)

Project description

There are a number of open questions in the context of dimer models on surfaces. The first two problems are the following: There is a notion of geometric exchange on Postnikov diagrams. It is a local move, defined on a region with four sides (strands). Consider the exchange graph whose vertices are the Postnikov diagrams on n strands on a disc, with an edge between two diagrams if they are connected by a geometric exchange. By [Postnikov06], this exchange graph is connected. Any Postnikov diagram corresponds to a dimer (with the natural potential mention above). There exists a notion of mutation of a dimer model, obtained from the mutation of quivers with potential [Derksen-Weyman-Zelevinsky]. Mutation of quivers is more general in the sense that it can be done at arbitrary vertices. However, it is not clear how to define the exchange on regions with more than four sides. We want to explore in this direction and find a notion of geometric exchange valid on arbitrary regions.

Furthermore, [Baur-King-Marsh] use (k,n)-diagrams for discs and define similar diagrams for k=2 for arbitrary surfaces without punctures. One of the aims is to find an appropriate notion of a Postnikov diagram for an arbitrary surface, even allowing punctures. This will involve the choice of a permutation for every boundary component of the surface. We expect that we still can find associated Frobenius categories and thus provide combinatorial models for many families of cluster categories.