Theta Functions and Log Calabi Yau Varieties

几何学中的一个中心开放问题，从理论物理学中发展出来，是镜像对称猜想，它指出一类几何对象，在数学的许多分支中具有根本重要性，以及理论物理学，所谓的Calabi-Yau流形，自然成对，它们是“镜像的“，因为一个的一个几何方面等效于其对的完全不同的几何方面。 这个猜想很重要，不仅因为卡-丘流形出现在数学的许多不同部分，而且因为镜像方面将数学的两个分支（辛几何和复几何，它们测量两个不同的方面）以前所未有的方式结合起来。 也许最根本的问题是，给定一个这样的对象，人们如何获得它的镜像伙伴。建议的研究的主要重点是一个详细的程序，用于构建一个给定的卡-丘镜像。在此奖项下，PI将继续指导博士后和研究生的培训。此外，他还将指导高中生的数学。该奖项是基于PI的猜想（与Gross，Hacking和Siebert联合），即某些流形（具有最大边界的log Calabi Yaus，或接近模空间的大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簇的经典theta函数。我们的建议是构造这些theta函数，并使用它们来证明一个令人惊讶的关于极化Calabi Yau对模的简单猜想，并将PI的构造扩展到更高的维度（联合格罗斯，黑客，（Siebert）极化K3表面的模量空间的几何紧致化。该奖项反映了NSF的法定使命，并通过使用基金会的学术价值和更广泛的影响审查标准。

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.