Cohomology of Moduli Spaces

至少在上世纪，研究复代数曲线及其在族中的变化是代数几何的基础。近几十年来取得了巨大的进步，并发现了与理论物理和数学其他部分的意想不到但极其重要的联系。然而，尽管经过了150年的研究，我们对曲线模空间拓扑的理解仍然非常不完整。这个项目的目的是获得这些和相关的模空间的拓扑结构的新的理解，也相关的模空间的向量丛在一个变化的曲线，通过代表他们的意义上的几何不变的理论和应用方法开发在过去的二十年来研究这种一致性。

The study of complex algebraic curves, and how they vary in families, has been fundamental to algebraic geometry for at least the last century. Huge advances have been made in recent decades, and unexpected but extremely important links with theoretical physics and other parts of mathematics have been discovered. Nonetheless, in spite of 150 years of investigation, our understanding of the topology of the moduli spaces of curves is still very incomplete. The aim of this project is to gain new understanding of the topology of these and related moduli spaces, and also related moduli spaces of vector bundles over a varying curve, by representing them as quotients in the sense of geometric invariant theory and applying methods developed over the last twenty years for studying such quotients.