Theoretical work in Mirror Symmetry:
The methodological heart of the project is a set of mathematical ideas called Mirror Symmetry which arose originally in string theory. My team has made important theoretical advances here. Some of these are directly related to Fano classification (for example the paper Laurent Inversion, by Coates--Kasprzyk--Prince, which introduces new methods for constructing a Fano manifold from its mirror partner, and papers by Andrea Petracci which give important "no-go" results that exhibit the difficulty of constructing smooth Fano manifolds by starting with geometries that have sharp corners called singularities and smoothing out these singularities). Other advances, in particular those by Thomas Prince and Hülya Argüz, apply some of the most recent advances in the field, which were in part inspired by the Fano classification program, to more classical questions in geometry. Still others, such as my papers with Iritani and with Lutz--Shafi and Doran--Kalashnikov, introduce new ideas and methods to Mirror Symmetry itself.
Mirror partners to Fano manifolds:
One of the central theoretical outputs of the project is a clear characterisation of the objects, called Maximally Mutable Laurent Polynomials, that correspond to Fano manifolds under mirror symmetry. This is spelled out in a series of papers with Dr Alexander Kasprzyk and others: "Maximally Mutable Laurent Polynomials" in Proceedings of the Royal Society A; "Databases of Quantum Periods for Fano Manifolds" in Nature Scientific Data; "Mirror Symmetry, Laurent Inversion and the Classification of Q-Fano Threefolds" which is under review; and "Computation and Data in the Classification of Fano Varieties" in the Nankai Symposium on Mathematical Dialogues. Databases of these mirror partners are available on Zenodo and from the Graded Ring Database, which is the main data repository in this field.
Computational algebra:
In joint work with Dr Alexander Kasprzyk of the University of Nottingham, we developed open-source mathematical software for large-scale computations in geometry and algebra. This project is ongoing, and underlies all of the computational aspects of the program. Source code and documentation are available at
https://bitbucket.org/pcas/(odnośnik otworzy się w nowym oknie) and
https://bitbucket.org/fanosearch/(odnośnik otworzy się w nowym oknie)Data and computations:
The open-source software just described, together with theoretical advances by my PhD student Elana Kalashnikov, has allowed us to analyse a new class of Fano manifolds called quiver flag zero loci. Computations on the Imperial College HPC cluster -- which would have taken centuries of run-time on a single machine -- allowed us to completely classify a broad class of these shapes. In this way we discovered more than 100 new four-dimensional Fano manifolds.
