### Project 12: "Exchange graphs of triangulations"

*This project was running from 2015 until 2019.*

Principal investigator: Karin Baur

University of Graz, Austria.

*Mentor for*: Oh.

### DK Student

**Second phase of the doctoral program:****Lukas Andritsch**(Austria; October 2015 - February 2019)

*Email*: lukas.andritsch@edu.uni-graz.at

*Mentors*: Oswin Aichholzer, Daniel Smertnig.

*Thesis Title*: "Combinatorial aspects of tilings".

*PhD Defense*: February 05, 2019.

*Referees*: K. Baur, P. Martin (University of Leeds), O. Aichholzer (TU Graz)

*Examiners*: K. Baur, P. Martin (University of Leeds)

### Associated Students

**Second phase of the doctoral program:****Jordan McMahon**(New Zealand; October 2015 - September 2019)

*Email*: jordan.mcmahon@uni-graz.at

*Mentors*: Alfred Geroldinger, Dusko Bogdanic.

*Thesis Title*: "Idempotent Ideals and Higher Auslander-Reiten theory".

*PhD Defense*: September 30, 2019.

*Referees*: Karin Baur, Pierre-Guy Plamondon (Paris Sud), Peter Jørgensen (Newcastle, UK).

**Hannah Vogel**(USA; April 2015 - September 2016)

*Email*: hannah.vogel@uni-graz.at

*Mentors*: Michael Kerber, Birgit Vogtenhuber.

*Thesis Title*: "Asymptotic triangulations and cluster algebras".

*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.

Showcases for possible PhD themes can be found here.