Homological Mirror Symmetry and Categorical Linear Systems

范畴论是一种将数学结构研究形式化的代数方法。该项目奠定了范畴论中经典几何方法的基础。今天，在21世纪的第二个十年，现代几何和理论物理比以往任何时候都更加交织在一起。数学和物理学的思想正在加速融合，同时基本粒子物理学正处于大型强子对撞机（LHC）新实验结果带来的深刻革命的尖端。与此同时，还有大量的数学工作要做，以便为正在提出的新物理理论提供一个合适的框架。在这个项目中研究的几何对象是这样一个框架的基础：同调镜像对称是对偶和更高类别的数学实现，经典流形的类似物，是量子场论的数学基础。这个项目所基于的这些新的几何风格将继续在理论物理的未来发展中发挥基础性作用。所提出的方法是基于Seidel，Ein，Lazarsfeld，Mustata，Nakamaye，Popa和Budur的开创性工作。PI将进一步推测范畴乘子理想层与范畴的奥尔洛夫谱有关。发展K-演算并使其严格化，将为代数几何中的经典问题，包括射影簇的合理性问题的研究开辟新的天地。特别是，PI计划考虑一些关于圆锥束和四维三次体的非合理性的一百多年前的问题。其应用范围超出了代数几何的范畴。辛几何中的经典问题也将被研究-使用福谷范畴的不变量，人们可以尝试用相同的Seiberg -维滕不变量来区分辛流形。该方法将K-演算与Kahler-Einstein度量存在性研究中使用的所谓品尝配置联系起来。一个有趣的问题是找到奥尔洛夫谱和卡勒-爱因斯坦度规之间的联系。该项目的另一个方向是研究K-演算与物理学的联系，其出发点是将K -演算的单值数据解释为有限稳定条件。

Category theory is an algebraic approach that formalizes the study of mathematical structures. This project lays the foundations of classical geometric methods in category theory. Today, in the second decade of the 21st century, modern geometry and theoretical physics are more intertwined than ever before. The convergence of ideas from mathematics and physics is accelerating at the same time as elementary particle physics is on the cusp of a profound revolution to be brought about by the new experimental results coming out of the Large Hadron Collider (LHC). At the same time, a lot of mathematical work remains to be done to provide a suitable framework for the new physical theories that are being proposed. The geometric objects investigated in this project are the foundations for such a framework: homological mirror symmetry is the mathematical realization of dualities and higher categories, the analogues of classical manifolds, are the mathematical foundation for quantum field theories. These new flavors of geometry on which this project is based will continue to play a fundamental role in the future development of theoretical physics.The proposed approach is based on the pioneering works by Seidel, Ein, Lazarsfeld, Mustata, Nakamaye, Popa, and Budur. The PIs will go further and conjecture that the categorical multiplier ideal sheaf is related to the Orlov spectrum of the category. Developing K-calculus and making it rigorous the PIs will break new ground in studying classical questions in Algebraic Geometry, including questions of rationality of projective varieties. In particular the PIs plan to consider some more than hundred years old questions about nonrationality of conic bundles and four dimensional cubics. The applications go beyond the scope of Algebraic Geometry. Classical questions in Sympletcic Geometry will be studied as well - using invariants of Fukaya category one can try to distinguish symplectic manifolds with the same Seiberg - Witten invariants. The approach connects K-calculus with so called tasting configurations used in the study of the existence of Kahler-Einstein metrics. An intriguing question is that of finding a connection between Orlov spectra and the existence of Kahler-Einstein metrics. Another direction of the project is the investigation of the connection of K-calculus with physics, for which a starting point is the interpretation of monodromy data of the K - calculus as limited stability conditions.