### Research

Interested in cluster algebras and their links to the geometry of surfaces, working under the supervision of Pavel Tumarkin and John Parker.

#### Marked exchange graphs and cluster automorphisms

Gave a link between automorphisms of skew-symmetrizable cluster algebras and
automorphisms of their exchange matrices. Introduced a marking on the exchange
graph such that automorphisms fixing the marking correspond exactly to cluster
automorphisms.

Published: Electon. J. Comb.,
Preprint: link.

#### Minimal mutation-infinite quivers

In 2015 classified all minimal mutation-infinite quivers. The webpage
containing the full classification, as well as images of all minimal
mutation-infinite quivers can be found here.

Published: Exp. Math.,
Preprint: link.

#### Properties of minimal mutation-infinite quivers

Worked with Matt Mills to determine various properties of minimal
mutation-infinite quivers and the cluster algebras generated by them. We show
that all these quivers have maximal green sequences, that these quivers are
Louise and so their cluster algebras are equal to their upper cluster algebra,
and that in most cases different classes of these quivers belong to different
mutation classes.

Preprint: link.

### Seminars

- Computing talks
- Jointly organise a series of computing talks at Durham with Sam Fearn, aimed at PhD students but frequently relevant to staff members. See the seminar archives or the supplementary material (Durham login required).
- Gandalf
- Organise the Gandalf (Geometry and Algebra Forum) pure maths student seminars for the academic year 2015-16. See the seminar series website for more info.
- Geometry & Topology
- Invited to give a talk at the Geometry and Topology seminar series at
Durham on 28th Jan 2016. Spoke on "Mapping classes, clusters and combinatorics".

Rough notes: pdf

### Teaching

- 2015-16
- Tutor for the linear algebra taught by Herbert Gangl and Daniele Dorigoni, with three groups of 10 to 15 first year students.
- 2014-15
- Tutor for Linear algebra, with three groups of first year students.

### Conference talks

A list of all conferences attended is available here.

- Rome - October 2016
- Gave a talk at the "Joint Notre-Dame - La Sapienza Conference on Lie
theory and cluster algebras" (site).
**Maximal green sequences for minimal mutation-infinite quivers**

Maximal green sequences of quivers have been widely studied, with applications to quantum dilogarithm identites, to paths in scattering diagrams and oriented exchange graphs, and to BPS spectra of certain quantum field theories. There have been a number of recent results determining which quivers admit a maximal green sequence and we will discuss how these can be used to show that all minimal mutation-infinite quivers, those whose proper subquivers are all mutation-finite, have maximal green sequences. Joint work with Matthew Mills.Slides: pdf

- Münster - March 2016
- Gave a short talk at the "Cluster algebras and geometry" conference
(site).
**Mapping classes, clusters and combinatorics**

Triangulations of surfaces have a cluster structure where triangle flips correspond to mutations and the surface's mapping class group has recently been shown to be isomorphic to the group of cluster automorphisms preserving this structure. We will discuss the links between these groups and the group of exchange graph automorphisms, before generalising to the skew-symmetrizable setting. This work provides a combinatorial approach to studying mapping class groups using graph automorphism groups.Rough notes: pdf

- Leicester - June 2015
- Invited to give a talk at the "Workshop of Cluster algebras and finite
dimensional algebras"
(site).
**Minimal mutation-infinite quivers**

Quivers constructed from hyperbolic Coxeter simplices give examples of minimal mutation-infinite quivers, however these are not the only such quivers. We will classify minimal mutation-infinite quivers through a number of moves and link the representatives of the classes with the hyperbolic Coxeter simplices, plus exceptional classes which are not related to simplices. This classification leads to a procedure to check whether a given quiver is mutation-infinite without computing any part of its mutation class.