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将共同努力系统地阐述和理解同调镜像对称，并将其扩展到一般类型的簇和非交换簇。这将需要在同调代数、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理论对应。数学所能做的论证就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深层和复杂结构的情况下尤其重要，这些结构难以通过直接实验检测。与此同时，这一努力将使回答一些纯数学（几何）问题成为可能，其中一些问题已经公开了很长时间。合作将通过定期会议和讲习班进行，并通过促进该领域的主要专家和年轻数学家（研究生和博士后）之间的互动进行;将通过定期举办冬季和夏季学校和会议促进这一迅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 ofthe homological mirror symmetry conjecture. Auroux, Katzarkov,Kontsevich, Orlov and Seidel will lead a concerted effort to formulateand understand homological mirror symmetry in systematic manner, andextend 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 toinvestigate applications of mirror symmetry to classical problems inalgebraic geometry (for example studying the rationality of certainalgebraic varieties) and symplectic topology (in particular, Lagrangiansubmanifolds). From a wider perspective, the project aims to provide a mathematicalcounterpart to some recent advances in theoretical physics. Thecontribution that mathematics can make is to verify the soundness andconsistency of physical intuition, and to prepare the general groundon which further development can occur. This is particularly importantin those situations where developments in physics suggest the presenceof deep and complex structures, which are difficult to detect by directexperiment. At the same time, this effort will make it possible to answersome purely mathematical (geometric) questions, some of which have beenopen for a long time. The collaborative effort will be carried out through regular meetingsand workshops, and by fostering interaction between leading experts inthe field and younger mathematicians (graduate students and postdocs);dissemination of knowledge in this rapidly evolving area of mathematicswill be facilitated by regularly held winter and summer schools andconferences.