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，Orlov和Seidel将共同努力，系统地提出和理解同调镜像对称性，并将其推广到各种一般类型和非对易变体。这将需要在同调代数，非对易几何和辛几何方面的一些基础工作。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)中的经典问题上的应用。从更广泛的角度来看，该项目旨在为理论物理的一些最新进展提供一个数学上的对应。数学所能做出的贡献是验证物理直觉的合理性和一致性，并为进一步发展做好准备。这一点在物理学的发展表明存在深部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探测到。与此同时，这一努力将使回答一些纯粹的数学(几何)问题成为可能，其中一些问题已经开放了很长一段时间。将通过定期举行会议和讲习班，并通过促进该领域的主要专家与年轻数学家(研究生和博士后)之间的互动来开展协作工作；将通过定期举办的冬季和夏季学校和会议促进这一迅速发展的数学领域知识的传播。

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.