Moduli Spaces and Moduli Problems

模理论关注的是理解空间中所有可能的形状。例如，如果取一个矩形并确定相对的边，就得到一个圆柱体，进一步确定剩余的边就得到数学家所说的环面——甜甜圈的表面。从不同形状的矩形开始，最终得到不同的环面。此外，我们也可以从平行四边形开始做类似的识别，以获得更多的环面。可以证明，这样的环面在本质上是不同的——如果一个人住在一个环面的表面上，从远处看不到整个表面，他仍然可以分辨出用来建造环面的那种平行四边形。那么环面模空间就是所有可能的本质不同的环面形状的集合。令人惊讶的是，这个“形状集”本身就是一个很好的几何对象。这个项目处理类似类型的问题：对一些几何物体进行分类，并提供一些关于它们的进一步信息，并研究由此产生的模空间。该项目旨在更好地理解各种紧模空间的几何形状。与各种合作者一起，PI将构建复杂曲线模空间的模紧化，以及具有规定的零倍数的微分，紧化Teichmuller动力学中所谓的地层-相空间。实归一化亚纯微分将应用于限定给定度平面曲线顶点的数目。本计画也将研究三次三重模空间的各种紧化的双几何和同调性。最后，本课题将继续研究阿贝尔变体模空间上的同调和高维循环。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。

Moduli theory concerns itself with understanding all possible shapes the space can have. For example, if one takes a rectangle and identifies opposite edges, one obtains a cylinder, and further identifying the remaining edges gives what mathematicians call a torus - the surface of a donut. Starting with rectangles of different shapes one ends up with different tori. Furthermore, one can also do similar identifications starting with parallelograms, to get more tori. It can be proven that the such tori are intrinsically different - if one lives on the surface of a torus and cannot see the whole surface from far away, one can still distinguish the kind of parallelogram that was used to build the torus. The moduli space of tori is then the set of all possible shapes of tori that are intrinsically different. Amazingly enough this "set of shapes" itself is a nice geometric object. This project deals with posing problems of similar type: of classifying some geometric objects, together with some further information on them, and of studying the resulting moduli spaces.The project aims to understand better the geometry of various compact moduli spaces. Together with various collaborators, the PI will construct a modular compactification of the moduli space of complex curves together with a differential with prescribed multiplicities of zeroes, compactifying the so-called strata - phase spaces in Teichmuller dynamics. Real-normalized meromorphic differentials will be applied to bound the number of cusps of plane curves of a given degree. The project also will investigate the birational geometry and homology of various compactifications of the moduli spaces of cubic threefolds. Lastly the project will continue the investigations on homology and higher dimensional cycles on moduli spaces of abelian varieties.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.