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

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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.