Bridgeland stability on Fukaya categories of Calabi-Yau 2-folds

Two key ideas in mathematics are symmetry and classification.

Symmetry is ubiquitous in mathematics, and is the source of endless fascination and study. Many symmetries are well-known, for example the symmetries of a cube or sphere, but others are far more mysterious and their study has led to great mathematical advances. Mirror symmetry of Calabi-Yau manifolds has excited much research in mathematics (for example, in Algebraic Geometry and Symplectic Topology), and also in theoretical physics through String Theory, but in general remains poorly understood. Mirror symmetry involves relating the geometry of two Calabi-Yau manifolds: one aspect of the symmetry is called the "A-model" and the other is the "B-model". Whilst there have been advances in understanding the B-model, we seem to currently lack the tools to adequately tackle the A-model. Our research proposal aims to give a complete understanding of the A-model for Calabi-Yau 2-folds, which would be a major achievement.

Classification results enable us to describe a large family of mathematical objects that are typically hard to understand in a simpler manner. A typical strategy for classification results in geometry, going back at least to Riemann's Uniformisation Theorem, is to find a special representative for a given class of geometric objects. The challenge then is to determine whether such a special representative exists and, when it does, whether it is unique. In our setting, the special representatives are called special Lagrangians and their uniqueness is known, but the problem of finding them in a given class has proven to be very difficult, despite many attempts to solve it. Our proposal aims to solve this problem for special Lagrangians completely in the setting of Calabi-Yau 2-folds.

The proposed research will combine techniques from distinct areas of mathematics (Symplectic Topology and Geometric Analysis), and it is often the case that some of the most exciting breakthroughs in mathematics occur when different areas are brought together. The connections to further areas of mathematics and theoretical physics mean that the impact of the proposed research is likely to be far-reaching and inspire many new research directions which will have a profound effect on the field.

The primary impact of this proposal will be academic, with secondary impacts to the general public and school children through public engagement and outreach activities.

The academic impact will mainly be achieved through research articles, presentations at seminars and conferences, the organization of a workshop, and the training of a PDRA.

The research articles will be disseminated through the standard channels, first posted online so as to freely available and so that researchers are made aware of the findings as soon as possible, and then published, with an aim to publish in high-impact journals.

The entire research team will be involved in presentations at seminars and conferences, which will be invaluable experience particularly for the PDRA, and ensure that the outcomes of the research proposal are known to a wide audience.

The workshop will provide an invaluable forum for the presentation of the results of the research proposal, give the PDRA experience in organizing a workshop, and bring together researchers from the distinct communities with interest in the proposed research goals (i.e. from Symplectic Topology, Geometric Analysis, Algebraic Geometry and String Theory).

The Symplectic Geometry PDRA will be trained in techniques so they can work at the interface between Symplectic Topology and Geometric Analysis, which very few researchers are capable of doing despite the fantastic opportunities that can arise from being well versed in this area. This will put the PDRA in a key position to explore exciting new research directions, that will certainly help their academic career and, together with the outcomes of the research proposal, help build the UK's strength in both Symplectic Topology and Geometric Analysis.

The public engagement and outreach activities will build on the significant experience of Co-I Jason Lotay and will involve a combination of public talks, talks in schools, articles written for a lay audience, and Art and Maths workshops in collaboration between Professor Lotay and artist Lilah Fowler. There will opportunities for the PDRA to be involved in these activities, and they will receive guidance and encouragement from Professor Lotay.