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

代数几何研究的是代数变体，即由多项式方程系统定义的几何对象。这门学科的一个主要目标是对代数簇进行分类，涉及的问题包括如何确定何时可以使用代数函数将一个簇转换为另一个簇，或者如何构造具有高度约束几何性质的簇。在这些经典问题和现代工具之间发现了令人惊讶的联系，特别是对象的派生范畴和它们的模空间。该项目旨在进一步开发这些工具，以便在尚未解决的猜想上取得进展。通过会议、研讨会和指导机会，该项目还将培训这一领域的新一代数学家。该项目有三个相关的研究目标。第一种是利用奇点的非对易分解来证明关于凝聚层的派生范畴的结构结果，其动机是Bondal-Orlov和Kuznetsov将这些范畴与双曲几何联系起来的猜想。第二个目标是构造Bridgeland稳定性条件，并研究它们的模空间的几何，无论是在一般情况下还是在特殊情况下。第三个目标是将上述主题的进展应用于经典问题，如Hyperkahler品种的分类和三次四重的合理性问题。该奖项反映了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 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.