Research

Undirected graphs famously come in three families: finite-type, affine-type, and wild-type. In my research, I am interested in the graphs that are “barely” wild—these are hyperbolic graphs, AKA the simply laced hyperbolic Dynkin diagrams.

A lot of these occur as “overextended” ADE diagrams (with some exceptional cases), such as E10, which has particular significance in mathematical physics. Appropriately, the associated Cartan hyperbolic signature, that is, one negative eigenvalue and the rest positive. As such, much of the work I do lives naturally in hyperbolic space. In the future, I hope to generalise this work to all 238 hyperbolic Dynkin diagrams.

One of the leading open problems in algebraic geometry is understanding the manifold of Bridgeland stability conditions on the derived category of a Calabi–Yau variety. Two properties of interest for Stab are connectedness and contractability. One way to tackle this is to exhibit Stab as a covering space of a complex hyperplane arrangement, and this has been studied in various contexts through the work of Ikeda, Rota and Hirano–Wemyss. Recently, Anno,Bezrukavnikov, and Mirkovic introduced another approach to this problem via real variations of stability conditions in which a real slice of the complex manifold is constructed.

My work has been to demonstrate new examples of this idea, coming from Coxeter arrangements of the above graphs. Roughly speaking, this involves choosing a bounded t-structure for each alcove in the arrangement and relating neighbouring t-structures via tilts of the abelian hearts. 

To generalise the above work outside of the Coxeter setting, we move from the classical Tits cone arrangement to intersection arrangements, introduced by Iyama and Wemyss. We take a graph as before, but now shade a subset of its vertices. The intersection arrangement associated with this data lives in dimension the number of unshaded vertices minus one.

This is shown in the examples at the top of the page. Note the effect that changing the shaded vertices has in the E10 examples. We can use these arrangements to study tilting theory and stability conditions on contracted preprojective algebras. I have been working to classify all tilings of the hyperbolic plane that arise as intersection arrangements of hyperbolic graphs. The pictures are all drawn using Mathematica with root system data computed in Magma, the code for which is available in my thesis.

In a collaboration with Pavel Shlykov (Glasgow), we classify Nakajima quiver varieties in occuring in dimension four, using a combinatorial method based on the balancing of weighted graphs. These come in three main "types", based on the nature of the unbalanced nodes, in the sense that if we double this weight and subtract all the neighbouring weights we obtain a negative number. 

To the right are some dimension vectors that give rise to four-dimensional quiver varieties.  They are all non-isomorphic as affine varieties.