FRG Collaborative Research: Homological Mirror Symmetry and its applications

FRG合作研究：同调镜像对称及其应用

• 批准号: 0652630
• 负责人: Denis Auroux
• 金额: $ 30万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-15 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652630&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Homological; Mirror

The main aim of this collaborative project is an in-depth study of the homological mirror symmetry conjecture. Auroux, Katzarkov, Kontsevich, Orlov and Seidel will lead a concerted effort to formulate and understand homological mirror symmetry in systematic manner, and extend it to varieties of general type and to noncommutative varieties. This will require some foundational work in homological algebra, noncommutative geometry, and symplectic geometry. Another goal is to investigate applications of mirror symmetry to classical problems in algebraic geometry (for example studying the rationality of certain algebraic varieties) and symplectic topology (in particular, Lagrangian submanifolds). From a wider perspective, the project aims to provide a mathematical counterpart to some recent advances in theoretical physics. The contribution that mathematics can make is to verify the soundness and consistency of physical intuition, and to prepare the general ground on which further development can occur. This is particularly important in those situations where developments in physics suggest the presence of deep and complex structures, which are difficult to detect by direct experiment. At the same time, this effort will make it possible to answer some purely mathematical (geometric) questions, some of which have been open for a long time. The collaborative effort will be carried out through regular meetings and workshops, and by fostering interaction between leading experts in the field and younger mathematicians (graduate students and postdocs); dissemination of knowledge in this rapidly evolving area of mathematics will be facilitated by regularly held winter and summer schools and conferences.

这个协作项目的主要目的是对同源镜对称性猜想的深入研究。 Auroux，Katzarkov，Kontsevich，Orlov和Seidel将领导以系统的方式制定和理解同源镜对称性的共同努力，并将其扩展到一般类型和非交通性品种的品种。这将需要在同源代数，非交通性几何形状和符号几何形状中进行一些基础工作。另一个目标是研究镜像在代数几何形状（例如研究某些代数品种的合理性）和符号拓扑（尤其是Lagrangian Submanifolds）中的经典问题的应用。从更广泛的角度来看，该项目旨在与理论物理学的一些最新进步提供数学上的对应。数学可以做出的贡献是验证物理直觉的健全性和一致性，并准备进一步发展的一般理由。这在物理学的发展表明存在深层和复杂结构的情况下尤其重要，而这些结构很难通过直接实验来检测。同时，这项工作将使回答一些纯粹的数学（几何）问题是可能的，其中一些问题已经开放了很长时间。合作努力将通过常规会议和研讨会以及培养该领域的主要专家与年轻数学家（研究生和博士学位）之间的互动来进行；定期举行的冬季和暑期学校和会议将促进知识的传播。