K3 Surfaces, Derived Categories, and Cubic Fourfolds

代数几何在现代数学中占据中心地位，与数论，表示论，同伦理论，复分析，数学物理和其他领域相互作用。 PI使用导出范畴研究代数几何中的具体问题，导出范畴最早出现在20世纪60年代，作为同调代数的簿记工具，但在最近几十年中，它已经成为一个基本的几何不变量，并作为从弦理论到代数几何的思想管道。该项目包含一个强有力的组成部分，旨在培训不同层次的学生在代数几何。研究计划包括三个相关的项目。 第一个目的是统一一些例子的“K3类别”，有时称为“非交换K3曲面”，出现在连接双有理和hyperkaehler几何，在一个共同的建设。 第二个目的是证明导出的Torelli定理对Calabi-Yau三重不成立，使用称为BCOV挠的不变量来区分具有相同Hodge结构的两个空间。 第三个奖项涉及关于四维立方超曲面的几个更经典的问题。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

Algebraic geometry occupies a central place in modern mathematics, interacting with number theory, representation theory, homotopy theory, complex analysis, mathematical physics, and other fields. The PI studies concrete problems in algebraic geometry using the derived category, which first appeared in the 1960s as a bookkeeping device for homological algebra but which in recent decades has emerged as a fundamental geometric invariant in its own right, and as a conduit of ideas from string theory to algebraic geometry. This project contains a strong component aimed to the training of students at different levels in algebraic geometry.The research program consists of three related projects. The first aims to unify a number of examples of "K3 categories," sometimes billed as "non-commutated K3 surfaces," that arise in connection with birational and hyperkaehler geometry, in a common construction. The second aims to show that a derived Torelli theorem does not hold for Calabi-Yau threefolds, using an invariant called BCOV torsion to distinguish between two spaces with the same Hodge structure. The third deals with several questions of a more classical flavor about four-dimensional cubic hypersurfaces.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.