FRG Collaborative Research: Homological Mirror Symmetry and its applications

这个合作项目的主要目的是深入研究同调镜像对称猜想。Auroux，Katzarkov，Kontsevich，奥尔洛夫和Seidel将领导一个协调一致的努力，制定和理解同调镜像对称系统的方式，并将其扩展到一般类型的品种和非交换品种。这需要在同调代数、非交换几何和辛几何方面做一些基础工作。另一个目标是研究镜像对称在代数几何（例如研究某些代数簇的合理性）和辛拓扑（特别是拉格朗日子流形）中的经典问题中的应用。从更广泛的角度来看，该项目旨在为理论物理学的一些最新进展提供数学对应。数学所能做的贡献是证实物理直观的可靠性和一致性，并为进一步的发展奠定基础。这在物理学的发展表明存在难以通过直接实验检测的深层和复杂结构的情况下尤其重要。与此同时，这一努力将使人们有可能回答一些纯粹的数学（几何）问题，其中一些已经开放了很长一段时间。将通过定期会议和讲习班开展合作，并促进该领域领先专家与年轻数学家（研究生和博士后）之间的互动;定期举办的冬季和夏季学校和会议将促进这一快速发展的数学领域的知识传播。

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.