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，系统地阐述和理解同调镜像对称，并将其推广到一般类型的变种和非对易变种。这需要在同调代数、非对易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，该项目旨在提供一个与理论物理最近的一些进展相对应的数学模型。数学可以做出的贡献是验证物理直觉的合理性和一致性，并为进一步发展奠定一般基础。这在那些物理学的发展表明存在深而复杂的结构的情况下尤其重要，而这些结构很难通过直接实验来探测到。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)之间的互动来开展合作工作；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 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.