Theta Functions and Log Calabi Yau Varieties

源于理论物理的几何学中的一个中心开放问题是镜像对称猜想，该猜想指出，一类在许多数学分支以及理论物理中具有基本重要性的几何对象，即所谓的Calabi-Yau流形，都是自然对，它们是“镜像”，因为一个对象的一个几何方面相当于其对的另一个完全不同的几何方面。这个猜想很重要，既是因为Calabi-Yau流形出现在数学的许多不同部分，也是因为镜像方面以以前没有预料到的方式将数学的两个分支(辛几何和复几何，衡量两个不同方面)联系在一起。也许最根本的问题是，给定一个这样的物体，一个人如何获得它的镜像伙伴。拟议研究的主要焦点是构建给定Calabi-Yau的镜像的详细计划。根据这一奖项，PI将继续指导博士后和研究生培训。此外，他还将在数学方面指导高中生。该奖项是基于Pi的猜想(与Gross，Hking和Siebert联合)，即某些流形(具有最大边界的对数Calabi Yaus，或接近模空间的大型复杂结构极限点的紧致Calabi Yaus)具有正则函数，极大地推广了极化阿贝尔变种的经典theta函数。我们的建议是构建这些theta函数，并用它们来证明关于极化Calabi Yau对的模数的一个令人惊讶的简单猜想，并将PI的构造(与Gross、Hking和Siebert一起)扩展到更高的维度，即极化K3表面的模空间的几何紧致。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the central open questions in geometry, growing out of theoretical physics, is the mirror symmetry conjecture which states that a class of geometric objects, of fundamental importance in many branches of mathematics, as well as theoretical physics, so called Calabi-Yau manifolds, come in natural pairs, which are "mirror " in that one geometric aspect of one is equivalent to a quite different geometric aspect of its pair. The conjecture is important both because Calabi-Yau manifolds appear in many diverse parts of mathematics, but also because the mirror aspect unites two branches of mathematics (symplectic and complex geometry, which measure the two different aspects) in ways not previously anticipated. Perhaps the most fundamental question is how, given one such object, one obtains its mirror partner. The main focus of the proposed research is a detailed program for constructing the mirror to a given Calabi-Yau. Under this award, the PI will continue mentoring postdocs and training of graduate students. In addition, he will also mentor high school students in mathematics.The award is based on the PI's conjecture (joint with Gross, Hacking, and Siebert) that certain manifolds (log Calabi Yaus with maximal boundary, or compact Calabi Yaus close to a the large complex structure limit point of the moduli space) come with canonical functions, vastly generalizing the classical theta functions for polarized Abelian varieties. The proposal is to construct these theta functions and use them to prove a surprisingly simple conjecture on moduli of polarized Calabi Yau pairs, and to extend to higher dimensions the PI's construction (joint with Gross, Hacking, and Siebert) of a geometric compactification of the moduli space of polarised K3 surfaces.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.