Moduli Spaces and Moduli Problems

模量理论关注的是理解空间可能具有的所有可能形状。例如，如果一个矩形和标识相反的边缘，一个得到一个圆柱体，并进一步标识其余的边缘给出数学家所谓的环面-一个甜甜圈的表面。从不同形状的矩形开始，以不同的环面结束。此外，也可以从平行四边形开始进行类似的识别，以获得更多的环面。可以证明的是，这样的环面在本质上是不同的-如果一个人生活在环面的表面上，不能从远处看到整个表面，他仍然可以分辨出用于建造环面的是什么样的环。环面的模空间则是本质上不同的环面的所有可能形状的集合。令人惊讶的是，这组“形状”本身就是一个很好的几何对象。该项目处理类似类型的问题：对一些几何对象进行分类，以及关于它们的一些进一步信息，并研究由此产生的模空间。该项目旨在更好地理解各种紧致模空间的几何。PI将与各种合作者一起构建复杂曲线的模空间的模紧化，以及具有规定多重性的零的微分，紧化Teichmuller动力学中所谓的层相空间。实正规化亚纯微分将被用来限制给定次数的平面曲线的尖点个数。该项目还将研究三次三重模空间的各种紧化的双有理几何和同调。最后，该项目将继续调查的同源性和高维循环的模空间的阿贝尔品种。这个奖项反映了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。

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.