EXC 2044 - T02: Moduli spaces in arithmetic and geometry

Basic data for this project

Description

The term “moduli space” was coined by Riemann for the space Mg parametrizing all one-dimensional complex manifolds of genus g. Variants of this appear in several mathematical disciplines. In algebraic geometry, Mg is constructed using geometric invariant theory and has a well-studied compactification introduced by Deligne and  Mumford. Through Teichmüller theory, these spaces are seen as objects of differential geometry, leading to useful cell decompositions and stratifications for computing intersection numbers, with interpretations in quantum field theory, as shown by Witten and Kontsevich. Moreover, Mg can be viewed as the classifying space of the diffeomorphism group of a surface or as a space of surfaces, a perspective used by Madsen and Weiss in proving the Mumford conjecture. Moduli spaces that are vast generalisations of Riemann’s concept are central for many research directions. In arithmetic geometry, Shimura varieties or moduli spaces of shtukas play an important role in the realisation of Langlands correspondences. Diffeomorphism groups of high-dimensional manifolds and moduli spaces of manifolds and of metrics of positive scalar curvature are studied in differential topology. Moduli spaces are also one of the central topics in our research in mathematical physics, where we study moduli spaces of stable curves and of Strebel differentials.