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，人们考虑了这些形状上的潜在函数和力场（微分形式），从爱因斯坦关于弯曲时空的物理学方法中得到了启示。力场是线性对象，因此可以用线性代数和群论的方法来研究。从这个角度来看，比较复杂的方法之一是模块形式。该项目将探索模块化形式对光滑非线性形状的约束，以及当形状获得奇点时，哪些特征会持续存在。为了描述奇异形状，我们必须引入更通用的工具，称为准模和模拟模形式。该项目为本科生和研究生研究人员提供了几种参与途径，并与辛几何和数论的邻近领域建立了富有成效的联系。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，扩展到枚举几何、镜像对称和Severi变种。这些领域通过模块化形式的出现及其概括（Siegel、quasi和mock）联系在一起。一个广泛的主题是k -平凡变量的重要性，最显著的是椭圆曲线。研究者将广泛推广关于局部对称空间上环值模形式的Borcherds定理，然后探索一系列几何含义。在志村完备变体和稳定地图的Kontsevich空间之间有一个有趣的类比。一个新的椭圆曲面的Torelli定理给出了一个度和格交换的猜想对应系统。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。

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.