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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，但其他对称性则更加神秘，对它们的研究导致了数学的巨大进步。卡-丘流形的镜像对称性激发了数学（例如代数几何和辛拓扑）以及理论物理学（通过弦理论）的许多研究，但总体上仍然知之甚少。镜像对称涉及到两个卡-丘流形的几何关系：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足够的工具来处理A模型。我们的研究计划旨在全面理解Calabi-Yau 2-folds的A模型，这将是一项重大成就。分类结果使我们能够以更简单的方式描述一大系列通常难以理解的数学对象。一个典型的战略分类结果的几何，至少可以追溯到黎曼的均匀化定理，是找到一个特殊的代表性为一类给定的几何对象。因此，面临的挑战是确定是否存在这样一位特别代表，如果存在，确定其是否独一无二。在我们的设置中，特殊代表被称为特殊拉格朗日量，并且它们的唯一性是已知的，但是在给定类中找到它们的问题已经被证明是非常困难的，尽管有许多尝试来解决这个问题。我们的建议旨在解决这个问题的特殊拉格朗日完全设置的卡-丘2倍。拟议的研究将联合收割机技术从不同的数学领域（辛拓扑和几何分析），而且通常情况下，数学中一些最令人兴奋的突破发生在不同领域结合在一起时。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，并激发了许多新的研究方向，这将对该领域产生深远的影响。