The Geometry of Quasi-modular Forms

该奖项支持代数几何领域的研究，该领域涉及由多变量多项式方程定义的非线性形状。为了更好地理解它们的几何形状，人们考虑这些形状上的势函数和力场（微分形式），从爱因斯坦对弯曲时空的物理学方法中得到启示。力场是线性对象，因此可以使用线性代数和群论的方法进行研究。从这个观点来看，更复杂的方法是模块化形式。这个项目将探讨从模块化形式放置在光滑的非线性形状的约束，然后保持形状时，获得奇点的功能。为了描述奇异形状，我们必须引入更通用的工具，称为拟模和模拟模形式。该项目有几个途径参与本科生和研究生的研究人员，以及富有成效的连接与辛几何和数论的邻近领域。该项目解决了复杂代数几何几个领域的核心问题，从霍奇理论开始，延伸到枚举几何，镜像对称和塞维里品种。这些领域通过模形式的出现和它们的推广联系在一起：Siegel，quasi和mock。一个广泛的主题是K-平凡品种的重要性，最显着的椭圆曲线。研究者将极大地推广Borcherds关于局部对称空间上的循环值模形式的定理，然后探索一系列几何含义。在完备的志村簇和稳定映射的Kontsevich空间之间有一个有趣的类比。一个新的Torelli定理的椭圆曲面导致一个系统的代数对应的交换度和generation.This奖项反映了NSF的法定使命，并已被认为是值得的支持，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。

This award supports research in the field of algebraic geometry, which deals with nonlinear shapes defined by polynomial equations in many variables. To better understand their geometry, one considers potential functions and force fields (differential forms) on these shapes, taking a cue from Einstein’s approach to physics on a curved spacetime. Force fields are linear objects, and thus can be studied using methods of linear algebra and group theory. Among the more sophisticated methods from this point of view are modular forms. This project will explore the constraints placed on smooth nonlinear shapes from modular forms, and then which features persist when the shapes acquire singularities. To describe singular shapes, we must introduce more general tools called quasi-modular and mock modular forms. The project has several avenues for participation by undergraduate and graduate student researchers, as well as fruitful connections with the neighboring fields of symplectic geometry and number theory. The project addresses central questions in several areas of complex algebraic geometry, beginning with Hodge theory and extending through enumerative geometry, mirror symmetry, and Severi varieties. These areas are tied together by the appearance of modular forms and their generalizations: Siegel, quasi, and mock. A broad theme is the importance of K-trivial varieties, most notably elliptic curves. The investigator will vastly generalize a theorem of Borcherds about cycle-valued modular forms on locally symmetric spaces, and then explore an array of geometric implications. There is an intriguing analogy between completed Shimura varieties and Kontsevich spaces of stable maps. A new Torelli theorem for elliptic surfaces leads to a system of conjectural correspondences which interchange degree and genus.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.