The Hilbert Scheme of Points in Higher Dimensions

世界上的模式既可以用几何空间来描述，也可以用代数方程来描述。如何从空间的定义方程中读出空间的几何性质是一个挑战。该项目将在概念和计算方面在这个问题上取得新的进展。作为该项目的一部分，研究生将被引入数学研究。点的Hilbert格式对给定代数变体的所有零维闭子格式进行分类。首席研究员和合作者将证明关于高维变化点的希尔伯特格式拓扑的第一个系统结果。在该项目的另一部分，首席研究员和研究生将通过一种新的探索性计算方法，在K3表面上使用计算机计算来探索Fano品种的分类。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

Patterns in the world can be described both by geometric spaces and by algebraic equations. It is a challenge to find how to read off the geometric properties of a space from its defining equations. The project will make new advances on the problem, in both conceptual and computational ways. As part of the project, graduate students will be introduced to mathematical research.The Hilbert scheme of points classifies all zero-dimensional closed sub-schemes of a given algebraic variety. The principal investigator and collaborators will prove the first systematic results about the topology of the Hilbert scheme of points on a high-dimensional variety. In another part of the project, the principal investigator and graduate students will explore the classification of Fano varieties by a new exploratory, computational method, using computer calculations on a K3 surface.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.