New Compactifications of Moduli Spaces of Curves

模空间是代数方程组的几何表示。代数、动力学和物理学中的许多基本问题都可以重新表述为模空间的几何问题，特别是曲线的模空间。本计画将对曲线的模空间的紧化进行系统的分类，并发展新的工具来研究这些紧化的几何。这些紧化之一已经位于连接拓扑学、组合学和量子场论的非凡网络的中心，并且很有可能这个网络适合于一个更全面的图景，这个图景平等地涉及所有的紧化。

A moduli space is a geometric representation of a system of algebraic equations. Many fundamental problems in algebra, dynamics and physics can be reformulated as questions concerning the geometry of moduli spaces, in particular the moduli space of curves. This project will produce a systematic classification of compactifications of the moduli space of curves, and develop new tools for investigating the geometry of these compactifications. One of these compactifications already lies at the centre of an extraordinary web of connections linking together topology, combinatorics and quantum field theory, and there is a strong possibility that this web fits into a more comprehensive picture involving all compactifications on an equal footing.