Cohomology of Moduli Spaces

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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.