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.

代数几何是对多项式方程（称为代数簇）解空间的研究。该学科的中心目标是代数簇的分类，涉及诸如如何确定何时可以将一个簇转化为另一个簇，或者如何构造具有指定几何性质的簇之类的问题。在这一追求中，一个反复出现的主题是使用代数不变量将问题转化为更容易处理的问题，例如上同调（测量空间中的“洞”）。该项目侧重于使用更精致的不变量，即派生范畴，它为代数簇几何提供了一个强大的窗口，并且还与其他学科（例如辛几何、表示论和理论物理学）相联系。该项目包括通过研讨会、讲习班和其他活动为该领域的学生和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思想之上，即派生类别应该通过将其分解成更小的部分（称为半正交分量）来研究，这些更小的部分可以被有效地视为非交换代数簇。首先，PI 将为这些非交换簇开发双有理几何工具，以期得到应用。其次，PI 将利用非交换簇的霍奇理论，在有关周期、布劳尔群和法诺簇的开放问题上取得进展。第三，PI 将研究 hyperkaehler 变体及其派生类别在非交换 K3 曲面方面的猜想描述。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。