Complex Geometric Properties of Period Maps

代数几何学研究多项式方程组的解的集合的几何性质。许多熟悉的集合都是以这种方式出现的，包括线和平面，以及圆和球体/肥皂泡。代数几何在代数学之外也有许多应用，包括统计学、机器人学、纠错码、遗传学、博弈论和整数规划。 “模问题”是代数几何的一个分支，研究这些集合的族。例如，可以考虑所有通过一个固定点的直线。这些家庭以及相关的问题可能相当复杂。他们研究的一个重要工具是“时期图”，这是一个与家庭有关的功能。周期图允许人们从数学的其他领域，包括复杂几何（定义函数不是多项式，如代数几何，而是全纯函数）和表示论（线性代数的复杂扩展）。该奖项支持：（i）研究周期图的性质，这些性质是由模问题的应用所激发的，以及（ii）培养研究生和早期职业博士后研究人员。PI的工作包括通过多个系和大学项目积极指导本科生和研究生。一半的项目旨在构建周期图的完成，其动机是模空间的紧致化的预期应用。 这包括一个Ohsawa-Takegoshi型扩张问题，它将完成一个程序来推广Satake-Bachelor-Borel紧化;对一个特定的Calabi-Yau簇族的这种构造的研究;以及对各种周期图的Kato-Bachelor型完备化的构造，包括与一个特定的Calabi-Yau 3-folds族相关的周期图。 其他项目继续由PI发起的一个计划，通过其特征形式（与周期图相关的无穷小，复几何不变量）研究周期图（和霍奇结构的变化）。 该项目的灵感来自于非常成功的Hwang-Mok项目，该项目通过各种最小有理切线来研究Fano流形。该奖项反映了NSF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而被认为值得支持。

Algebraic geometry studies geometric properties of sets that arise as solutions to systems of polynomial equations. Many familiar sets arise in this way, including lines and planes, and circles and spheres/soap bubbles. Algebraic geometry also has many applications outside of mathemayics, including statistics, robotics, error-correcting codes, phylogenetics, game theory and integer programming. “Moduli problems” is a sub-branch of algebraic geometry that studies these sets in families. For example, one might consider all the lines passing through a fixed point. These families, and the related questions, can be quite complicated. An important tool in their study is the “period map”, a function that is associated with the family. The period map allows one to bring to bear techniques from other areas of mathematics, including complex geometry (where the defining functions are not polynomials, as in algebraic geometry, but holomorphic functions) and representation theory (a sophisticated extension of linear algebra). This award supports both: (i) research into properties of period maps that are motived by applications to moduli problems, and (ii) the training of graduate students and early career postdoctoral researchers. The PI’s work includes active mentorship of both undergraduate and graduate students through multiple departmental and university programs.Half of the projects aim to construct completions of period maps, motivated by anticipated applications to compactifications of moduli spaces. This includes an Ohsawa-Takegoshi type extension problem that will complete a program to generalize the Satake-Baily-Borel compactification; an investigation of this construction for a specific family of Calabi-Yau varieties; and the construction of a Kato-Usui type completions for various period maps, including that associated to a specific family of Calabi-Yau 3-folds. The other projects continue a program initiated by the PI to study period maps (and variations of Hodge structure) by their characteristic forms (which are infinitesimal, complex-geometric invariants associated with a period maps). This program is inspired by a close analogy with the very successful Hwang-Mok program to study Fano manifolds via their varieties of minimal rational tangents.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.