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Gromov-Witten Theory: Mirror Symmetry, Birational Geometry, and the Classification of Fano Manifolds

Periodic Reporting for period 4 - GWT (Gromov-Witten Theory: Mirror Symmetry, Birational Geometry, and the Classification of Fano Manifolds)

Periodo di rendicontazione: 2021-04-01 al 2022-09-30

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.

Outputs from this project include databases of Fano manifolds in dimension four, many of which are new, as well as a suite of software products that enable computer algebra calculations on a massive scale. One of the most surprising discoveries during the project -- and this was not anticipated in the project proposal, but rather was discovered during some unplanned experiments during the COVID lockdown period when we were unable to continue with our planned work -- is that we can use tools from data science and machine learning to find new Fano manifolds, and to predict geometric properties of those manifolds that are otherwise hard to compute.
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/ 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 project has now ended, so the results are summarised above.

For the next section, the system only allows me to enter one URL but there are several associated with the project, as follows:
the project is described on my website coates.ma.ic.ac.uk
databases are available at grdb.co.uk
open source mathematical software can be found at https://bitbucket.org/pcas/ and https://bitbucket.org/fanosearch/
A K3 surface obtained by taking a slice through the three-dimensional Fano manifold V6.
Some of my research team at the Imperial Lates XMaths outreach event.
A K3 surface obtained by taking a slice through the three-dimensional Fano manifold MM3-48.
A K3 surface obtained by taking a slice through the three-dimensional Fano manifold MM2-30.
Discussing with Prof. Bourguignon at the Imperial ERC celebration.