## Periodic Reporting for period 3 - GWT (Gromov-Witten Theory: Mirror Symmetry, Birational Geometry, and the Classification of Fano Manifolds)

Reporting period: 2019-10-01 to 2021-03-31

Fano manifolds are basic building blocks in geometry -- they are, in a precise sense, atomic pieces of mathematical shapes. The central objective of this project is to find and classify Fano manifolds in dimensional four, using ideas that come from string theory in theoretical physics in combination with large-scale computational algebra calculations on HPC clusters. One can think of this as building a "periodic table" of mathematical shapes. This is fundamental blue-sky research, with applications in other areas of mathematics, in physics, and in scientific computation. At this half-way point of the project we are making good progress on the theoretical aspects of the project, but are a little behind schedule in the computational aspects. Nonetheless, overall progress is satisfactory -- in particular, we have discovered more than 100 new four-dimensional Fano manifolds.

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 paper with Iritani, introduce new ideas and methods to Mirror Symmetry itself.

Computational algebra:

In joint work with Dr Alexander Kasprzyk of the University of Nottingham, we are developing 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/ and https://bitbucket.org/fanosearch/

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.

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 paper with Iritani, introduce new ideas and methods to Mirror Symmetry itself.

Computational algebra:

In joint work with Dr Alexander Kasprzyk of the University of Nottingham, we are developing 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/ and https://bitbucket.org/fanosearch/

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.

By the end of the project, we expect to have:

* a sound theoretical characterisation of the objects ("maximally mutable Laurent polynomials") that correspond to Fano manifolds, of any dimension, under Mirror Symmetry;

* complete, publicly-available databases of Fano manifolds in dimension 4 and Fano orbifolds in dimensions 3 and 4;

* a rigorous proof that our classifications are complete;

* a freely-available open-source computational algebra system for cluster-scale computations in geometry and algebra.

* a sound theoretical characterisation of the objects ("maximally mutable Laurent polynomials") that correspond to Fano manifolds, of any dimension, under Mirror Symmetry;

* complete, publicly-available databases of Fano manifolds in dimension 4 and Fano orbifolds in dimensions 3 and 4;

* a rigorous proof that our classifications are complete;

* a freely-available open-source computational algebra system for cluster-scale computations in geometry and algebra.