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

Calabi-Yau 2 倍 Fukaya 类别上的 Bridgeland 稳定性

基本信息

批准号：
EP/T012749/1
负责人：
Dominic Joyce
金额：
$ 66.58万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT012749%2F1
关键词：
Bridgeland stability Fukaya categories Calabi

项目摘要

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.

数学的两个关键思想是对称性和分类。对称性在数学上无处不在，并且是无休止的迷恋和研究的根源。许多对称性是众所周知的，例如立方体或球体的对称性，但其他对称性则更加神秘，他们的研究导致了巨大的数学进步。卡拉比远流形的镜像对称性激发了许多在数学方面的研究（例如，在代数几何和象征性拓扑结构中），以及通过弦乐理论中的理论物理学研究，但一般而言仍然知之甚少。镜像对称性涉及关联两个卡拉比（Calabi-Yau）歧管的几何形状：对称的一个方面称为“ A模型”，另一个是“ B-模型”。尽管在理解B模型方面取得了进步，但我们目前似乎缺乏足够的工具来适应A模型。我们的研究建议旨在使Calabi-Yau的A模型完全了解2倍，这将是一个主要的成就。分类结果使我们能够描述一个通常很难以更简单方式理解的大型数学对象。分类的典型策略导致几何形状至少可以追溯到里曼的统一定理，是为给定类别的几何对象找到一个特殊的代表。那时的挑战是确定这种特殊的代表是否存在，并且当它确实存在时，它是否是唯一的。在我们的环境中，特别代表被称为特殊的Lagrangians，他们的独特性是已知的，但是在给定班级中找到它们的问题已被证明是非常困难的，尽管许多尝试解决了它。我们的建议旨在在2倍的Calabi-yau的环境中完全解决特殊拉格朗日人的问题。拟议的研究将结合来自不同数学领域（符号拓扑和几何分析）的技术，通常在不同领域中发生一些最令人兴奋的数学突破的情况是，这种情况通常是如此。与数学和理论物理学的进一步领域的联系意味着，拟议的研究的影响可能是深远的，并激发了许多新的研究方向，这将对该领域产生深远的影响。