Homological Mirror Symmetry and Categorical Linear Systems

同调镜像对称和分类线性系统

• 批准号: 1502162
• 负责人: Ludmil Katzarkov
• 金额: $ 17万
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502162&HistoricalAwards=false
• 关键词: Homological; Mirror; Symmetry; Categorical; Linear

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.

类别理论是一种代数方法，它正式地研究了数学结构的研究。该项目在类别理论中奠定了经典几何方法的基础。如今，在21世纪的第二个十年中，现代的几何形状和理论物理学比以往任何时候都更加交织。来自数学和物理学的思想的融合正在同时加速，因为基本粒子物理学正是由大型强子对撞机（LHC）产生的新实验结果带来的深刻革命的风口。同时，为了为正在提出的新物理理论提供合适的框架还有许多数学工作要做。该项目中研究的几何对象是这样一个框架的基础：同源镜对称性是二元性的数学实现，而更高的类别是经典歧管的类似物，是量子场理论的数学基础。该项目所基于的这些几何形状的这些新口味将继续在理论物理的未来发展中发挥基本作用。拟议的方法基于Seidel，Ein，Ein，Lazarsfeld，Mustata，Mustata，Nakamaye，Popa和Budur的开创性作品。 PI将进一步猜测，分类乘数理想的支架与该类别的Orlov频谱有关。开发K钙库并使PI的严格性将在研究代数几何学中研究经典问题，包括投射品种合理性的问题。特别是，PIS计划考虑有关圆锥捆绑包和四个立方体的非理性的数百年以上的问题。应用超出了代数几何形状的范围。还将研究符号几何形状中的经典问题 - 使用福卡亚类别的不变性，可以尝试以相同的seiberg -witten不变性来区分符号歧管。该方法将K-Calculus与Kahler-Einstein指标存在的所谓品尝构型联系起来。一个有趣的问题是找到Orlov光谱与Kahler-Einstein指标的存在之间的联系。该项目的另一个方向是研究K -Calculus与物理学的连接，其起点是将K -colculus的单片数据解释为有限的稳定性条件。