FRG: Collaborative Research: Derived Categories, Moduli Spaces, and Classical Algebraic Geometry

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Algebraic geometry is the study of algebraic varieties, the geometric objects defined by systems of polynomial equations. A driving goal of the subject is the classification of algebraic varieties, involving questions like how to determine when one variety can be transformed into another using algebraic functions, or how to construct varieties with highly constrained geometric properties. Surprising connections have been found between these classical problems and modern tools in the subject, especially derived categories and their moduli spaces of objects. This project aims to further develop these tools in order to make progress on outstanding conjectures. Through conferences, workshops, and mentoring opportunities, the project will also train a new generation of mathematicians in this area. The project has three related research goals. The first is to use noncommutative resolutions of singularities to prove structural results about derived categories of coherent sheaves, motivated by conjectures of Bondal-Orlov and Kuznetsov relating these categories to birational geometry. The second goal is to construct Bridgeland stability conditions and study the geometry of their moduli spaces, both in general settings and cases of special interest. The third goal is to apply advances on the above topics to classical problems, like the classification of hyperkahler varieties and the rationality problem for cubic fourfolds.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.