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

数学中的两个关键概念是对称性和分类。对称性在数学中无处不在，是无穷无尽的魅力和研究的源泉。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，他们的研究带来了巨大的数学进步。Calabi-Yau流形的镜像对称性在数学(例如，代数几何和辛拓扑)以及弦理论的理论物理中引起了许多研究，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”，另一个被称为“B模型”。虽然在理解B型模式方面取得了进展，但我们目前似乎缺乏充分处理A型模式的工具。我们的研究提案旨在让我们对Calabi-Yau 2折叠的A模型有一个完整的理解，这将是一项重大的成就。分类结果使我们能够描述一大类通常很难用更简单的方式理解的数学对象。在几何学中，一个典型的分类策略，至少可以追溯到黎曼的统一化定理，就是为给定的一类几何对象找到一个特殊的代表。那么，挑战就是确定这样一位特别代表是否存在，以及当它存在时，它是否独一无二。在我们的背景下，特殊代表被称为特殊拉格朗日，他们的独特性是已知的，但在给定的类别中找到他们的问题被证明是非常困难的，尽管有很多尝试来解决这个问题。我们的建议旨在在Calabi-Yau 2重的背景下完全解决特殊拉格朗日函数的这个问题。建议的研究将结合不同数学领域(辛拓扑和几何分析)的技术，并且通常情况下，当不同领域结合在一起时，数学中一些最令人兴奋的突破发生。与进一步的数学和理论物理领域的联系意味着，拟议的研究的影响可能是深远的，并激励许多新的研究方向，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.