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

数学中的两个关键思想是对称和分类。对称在数学中无处不在，是无穷魅力和研究的源泉。许多对称性是众所周知的，例如立方体或球体的对称性，但其他对称性要神秘得多，它们的研究导致了巨大的数学进步。Calabi-Yau流形的镜像对称性在数学（例如，代数几何和辛拓扑）以及通过弦理论在理论物理中激发了许多研究，但总体上仍然知之甚少。镜像对称涉及到两个Calabi-Yau流形的几何关系：对称的一个方面被称为“a模型”，另一个是“b模型”。虽然在理解b模型方面取得了进展，但我们目前似乎缺乏充分解决a模型的工具。我们的研究计划旨在完整地理解Calabi-Yau 2-fold的a模型，这将是一个重大的成就。分类结果使我们能够以更简单的方式描述一大类通常难以理解的数学对象。几何分类结果的一个典型策略，至少可以追溯到黎曼均匀化定理，是为给定的几何对象类别找到一个特殊的代表。因此，挑战在于确定这样一位特别代表是否存在，如果存在，是否独一无二。在我们的环境中，特别代表被称为特殊拉格朗日量，它们的独特性是已知的，但是在给定的类中找到它们的问题已被证明是非常困难的，尽管有许多尝试来解决它。我们的提议旨在完全解决Calabi-Yau 2-fold背景下的特殊拉格朗日量的这一问题。拟议的研究将结合不同数学领域的技术（辛拓扑和几何分析），通常情况下，当不同领域结合在一起时，一些最令人兴奋的数学突破就会发生。与数学和理论物理的进一步领域的联系意味着拟议研究的影响可能是深远的，并激发许多新的研究方向，这将对该领域产生深远的影响。

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.