CAREER: Geometry of Derived Categories

职业：派生类别的几何

• 批准号: 2143271
• 负责人: Alexander Perry
• 金额: $ 40万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143271&HistoricalAwards=false
• 关键词: CAREER; Geometry; Derived; Categories

Algebraic geometry is the study of the spaces of solutions to polynomial equations, known as algebraic varieties. A central goal of the subject is the classification of algebraic varieties, involving questions such as how to determine when one variety can be transformed into another, or how to construct varieties with specified geometric properties. In this pursuit, a recurring theme is to translate the problem into a more tractable one using an algebraic invariant, like cohomology (which measures the "holes" in a space). This project focuses on the use of a more refined invariant, the derived category, which provides a powerful window into the geometry of algebraic varieties, and also connects with other subjects, for example symplectic geometry, representation theory, and theoretical physics. This project includes training and research opportunities for students and early-career researchers in this area, through seminars, workshops, and other activities. In more detail, the project builds on the influential idea that derived categories should be studied by breaking them into smaller pieces, called semiorthogonal components, which can fruitfully be regarded as noncommutative algebraic varieties. First, the PI will develop tools from birational geometry for these noncommutative varieties, with a view toward applications. Second, the PI will exploit the Hodge theory of noncommutative varieties to make progress on open problems about cycles, Brauer groups, and Fano varieties. Third, the PI will study conjectural descriptions of hyperkaehler varieties and their derived categories in terms of noncommutative K3 surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数几何是研究多项式方程的解的空间，称为代数簇。该主题的一个中心目标是代数簇的分类，涉及的问题，如如何确定何时可以将一个品种转化为另一个品种，或如何构建具有指定几何性质的品种。在这一追求中，一个反复出现的主题是使用代数不变量（如上同调（测量空间中的“洞”））将问题转化为更容易处理的问题。这个项目的重点是使用一个更精细的不变量，派生类别，它提供了一个强大的窗口到代数簇的几何，也与其他学科，例如辛几何，表示论和理论物理连接。该项目包括通过研讨会、讲习班和其他活动为学生和这一领域的早期职业研究人员提供培训和研究机会。 更详细地说，该项目建立在一个有影响力的想法上，即派生范畴应该通过将它们分成更小的部分来研究，称为半正交分量，这些分量可以被有效地视为非交换代数簇。首先，PI将从双有理几何中开发用于这些非交换变体的工具，以期应用。第二，PI将利用霍奇理论的非交换品种取得进展的开放问题的周期，布劳尔群，和法诺品种。第三，PI将研究hyperkaehler品种及其衍生类别的非对易K3 surfaces.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。