Research interestsMy research focusses on problems in geometry related to wall-crossing phenomena observed in mathematical physics. In particular I am interested in spaces of stability conditions on triangulated categories, a geometric invariant of a triangulated category with walls along which the class of stable objects jumps. Most of the articles below can be found on my arXiv page.
A construction of Frobenius manifolds from stability conditions (2016, with Anna Barbieri and Jacopo Stoppa)
This article provides a construction of Frobenius manifold structures on spaces of stability conditions of CY3 categories via Donaldson-Thomas theory. We show that this construction when applied to the An quivers recovers Saito's Frobenius manifold structure on the unfolding spaces of An singularities.
Stability conditions and the A2 quiver (2014, with Tom Bridgeland and Qiu Yu)
This article gives a uniform description of a canonical connected component of the space of stability conditions on the Calabi-Yau categories of dimension at least 2 generated by a pair of spherical objects with a unique non-trivial extension between them (cf the article below in CY dimension 3). We give a biholomorphic map from the universal cover of the space of cubic polynomials with distinct roots to this connected component, lifting the period map of certain meromorphic differentials on the plane over cycles connecting the zeroes of the cubic polynomial. We show how this can be obtained by applying Dubrovin's almost duality to Saito's Frobenius manifold structure on the unfolding space of the A2 singularity, or, equivalently, the space of stability conditions of the bounded derived category of finite dimensional representations the A2 quiver.
This article describes a connected component of the space of stability conditions on the derived category of the category of dg modules with finite dimensional cohomology over the Ginzburg algebra of the A2 Dynkin diagram. This is the one of the simplest examples of the Calabi-Yau-3 triangulated categories associated to a marked bordered surface, categorifing the associated cluster algebras, whose spaces of stability conditions were subsequently considered by Bridgeland and Smith. In constrast to their approach via trajectories of quadratic differentials motivited by Gaiotto-Moore-Neitzke, we use some classical tools from projective geometry to exhibit this connected component as the universal cover of a C*-bundle over the moduli space of elliptic curves by considering the periods of a meromorphic differential on the universal family.
Stability conditions and Seiberg-Witten quivers (2014, PhD thesis, University of Sheffield)
This generalises the techniques of the above article on the A2 quiver to describe a connected component of the space of numerical stability conditions on the Calabi-Yau-3 triangulated categories associated to finite and affine Dynkin diagrams whose Cartan matrix is of rank 2. This class of triangulated categories arises in the geometric engineering of the asymptotically free SU(2) Seiberg-Witten theories and their degenerations into Argryes-Douglas theories. They are also in bijection with the Painlevé equations which define the isomonodromic deformations of the corresponding moduli spaces of flat connections studied in Gaiotto-Moore-Neitzke.
Etale cohomology (2008, Part III Essay, University of Cambridge)
This is an introduction to étale cohomology written whilst a was a Part III (Master's) student at the University of Cambridge. As with many Part III essays it was written in a very short space of time whilst the material on which it is based was all very new to me. I put it here not because it is a clear, well-written, error-free survey (it is not), but it might give some idea of what a Part III essay is like.