Moduli and Periods

代数几何学研究的是可以用代数定义的物体的几何性质。这样的对象，称为代数簇，是非常特殊的，有丰富的结构。因此，其他数学领域的技术，如算术，拓扑学和微分几何，在他们的研究中被采用。在代数几何中，这个项目研究模空间的几何。模空间是具有规定数值不变量的所有物体形状的空间。模空间的研究对于数学和理论物理的几个分支都是至关重要的。特别是，在过去的三十年里，数学和物理之间有很深的交叉影响，围绕着一类特殊的代数簇，卡-丘三重及其模。该项目的主要目标之一是开发用于研究卡-丘变体模空间的工具。本项目从周期映射的角度研究模空间。周期图是理解阿贝尔簇和K3曲面的模空间几何的重要工具。除了这些经典的情况下，调查的周期地图是困难得多，由于一些非平凡的和高度超越的条件（格里菲斯的横截性条件）所满足的霍奇结构的变化产生的几何设置。尽管如此，最近的进展，在该领域的一些非经典的情况下，特别是表面的一般类型的小不变量，和卡-丘三倍的调查成为可能。在另一个方向，几个问题的模空间的K3曲面和超Kaehler流形的几何形状进行了调查，通过周期maps.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Algebraic geometry is concerned with the study of geometric properties of objects that can be defined algebraically. Such objects, called algebraic varieties, are very special and have a rich structure. Consequently, techniques from other fields of mathematics, such as Arithmetic, Topology, and Differential Geometry, are employed in their study. Within algebraic geometry, this project studies the geometry of moduli spaces. A moduli space is the space of all shapes of objects with prescribed numerical invariants. The study of moduli spaces is of central importance to several branches of mathematics and theoretical physics. In particular, over the past three decades there were deep cross-influences between Mathematics and Physics centered around a special class of algebraic varieties, the Calabi-Yau threefolds, and their moduli. One of the main thrusts of this project is the development of tools for the study of moduli spaces of Calabi-Yau varieties. In this project, the moduli spaces are studied from the perspective of period maps. The period map is an essential tool for understanding the geometry of the moduli space of abelian varieties and K3 surfaces. Beyond these classical cases, the investigation of the period maps is much more difficult due to some non-trivial and highly transcendental conditions (the Griffiths' transversality conditions) satisfied by the Variations of Hodge Structure arising in the geometric setting. Nonetheless, recent progress in the field makes possible the investigation of some non-classical situations, especially surfaces of general type with small invariants, and Calabi-Yau threefolds. In a different direction, several questions about the geometry of the moduli space of K3 surfaces and hyper-Kaehler manifolds are investigated by means of period maps.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.