Some problems in derived algebraic geometry

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI建议将一般分类方法应用于代数几何中的具体问题。特别是PI提出了一个明确的概念，解决奇点。这种同调方法是一种将通常的交换代数几何与非交换代数几何统一起来的方法。PI还建议研究三角范畴理论中的基本问题：增强的存在性和唯一性。PI建议应用强大的抽象方法来研究现代几何和数学物理中的具体问题。特别是一个抽象的概念，决议的奇异性（“平滑”的空间）提出。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI proposes to apply general categorical methods to concrete problems in algebraic geometry. In particular the PI suggests a categorical notion of resolution of singularities. This homological approach is a way to unite the usual commutative algebraic geometry with the noncommutative one. The PI also proposes to investigate the basic problem in the theory of triangulated categories: the existence and uniqueness of enhancements.The PI suggests to apply powerful abstract methods to study concrete problems in modern geometry and mathematical physics. In particular an abstract notion of a resolution of singularities ("smoothing" of the space) is proposed.