Mirror symmetry

I worked with Alessio Corti and Al Kasprzyk at Imperial College, London in July-September 2014 funded by the LMS working on various problems involving orbifold del Pezzo surfaces: primarily...

  • reconstructing del Pezzos from a given Hilbert series, and the geometrical and combinatorial consequences such a classification has (for the realisability of a Hilbert series as the Hilbert series of an orbifold del Pezzo surface, and for the quasiperiod collapse of Fano polygons),

  • considering criteria for the existence of toric degenerations and examples where they do not exist, and

  • the structure present in a class of cyclic quotient surface singularities called residual singularities that were recently defined by Akhtar and Kasprzyk here. These are the \(\mathbb{Q}\)-Gorenstein 'unsmoothable' parts of a quotient singularity and hence are central to deformation theoretic and mirror symmetric questions posed about orbifold surfaces. By the end of my time at Imperial I received the informal title "resident expert on residual surface singularities".

    I would like to thank all of the members of the Fano group at Imperial for the welcoming and highly collaborative environment with which I was presented.

    Preprints

  • Reconstruction of orbifold del Pezzo surfaces from Hilbert series

  • A geometrical explanation of quasi-period collapse

  • Contributions from quotient singularities on del Pezzo surfaces.

    Posters

  • BrAG 2016.

    I am becoming increasingly interested in cluster algebras both intrinsically and for the role they play in mirror symmetry. See here for a note on how finite-type cluster algebras fit into the network of Dynkin correspondences.

    Preprints

  • Relative realisations of cluster algebras

  • Pythagorean triples and cluster algebras

  • Reconciling mutations (mostly survey plus some goodies).

    Research.