Moduli and Periods

代数几何是研究可以用代数定义的物体的几何性质的学科。这样的对象被称为代数变量，它们非常特殊，结构丰富。因此，从其他数学领域的技术，如算术，拓扑学和微分几何，被用于他们的研究。在代数几何中，这个项目研究模空间的几何。模空间是具有规定的数值不变量的物体的所有形状的空间。模空间的研究对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，数学和物理之间产生了深刻的交叉影响，其核心是一类特殊的代数变体，即Calabi-Yau三倍函数及其模。该项目的主要目标之一是开发用于研究卡拉比-丘品种模空间的工具。在这个项目中，模空间是从时期地图的角度来研究的。周期映射是理解阿贝尔变体和K3曲面的模空间几何的重要工具。除了这些经典案例之外，由于几何环境中出现的Hodge结构的变化所满足的一些非平凡的和高度超越的条件（Griffiths'横截性条件），对周期图的研究要困难得多。尽管如此，该领域的最新进展使得研究一些非经典情况成为可能，特别是具有小不变量的一般型曲面和Calabi-Yau三倍曲面。在另一个方向上，利用周期映射研究了关于K3曲面和超kaehler流形模空间几何的几个问题。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。

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.