CAREER: Geometry of Derived Categories

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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.