CAREER: Geometry of Derived Categories

代数几何是研究多项式方程的解的空间，称为代数簇。这门学科的一个中心目标是对代数簇进行分类，涉及的问题包括如何确定一个簇何时可以转换为另一个簇，或者如何构造具有特定几何性质的簇。在这种追求中，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，如上同调(它测量空间中的“洞”)。这个项目专注于使用一种更精细的不变量，派生范畴，它提供了一个了解代数簇几何的强大窗口，也与其他学科联系在一起，例如辛几何、表示理论和理论物理。该项目包括通过研讨会、讲习班和其他活动，为这一领域的学生和早期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之上，即应该通过将派生范畴分解成更小的部分来研究它们，这些部分被称为半正交分量，可以有效地被视为非交换代数簇。首先，PI将从二次几何为这些非对易变种开发工具，以期应用。其次，PI将利用非对易变元的Hodge理论在关于圈、Brauer群和Fano变元的公开问题上取得进展。第三，PI将根据非交换K3曲面研究Hyperkaehler变种及其派生类别的猜想描述。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.