Cohomology of Moduli Spaces

研究复杂的代数曲线，以及它们如何在家族中变化，至少在上个世纪一直是代数几何的基础。近几十年来取得了巨大的进步，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，我们对曲线模空间的拓扑学的认识仍然很不完整。这个项目的目的是通过将这些空间和相关的模空间表示为几何不变理论意义上的商，并应用过去二十年发展起来的研究这些商的方法，来获得对这些空间和相关的模空间以及变化曲线上的向量丛的相关的模空间的新的理解。

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.