This topic course breaks into two topics: 1. On moduli space of algebraic curves and mapping class groups. 2. On the structure of classical diffeomorphism groups
1. The moduli space of curves is a central topic in low dimensional topology, algebraic geometry and complex analysis. It is both the moduli space of algebraic curves, hyperbolic surfaces and Riemann surfaces. It is also virtually the classifying space of surface bundles. Thus, the study of the moduli spaces always has many different angles.
In this course, we will present some of Harer’s result on the comparison of moduli space with arithmetic groups. For example, we will prove high connectivity of curve complex, Harer stability, computation of Euler characteristic of moduli space. We will introduce Fenchel-Nielsen coordinate and a magic formula of Wolpert and draw connection with Mirzakhani’s work on counting simple closed curves on hyperbolic surfaces.
2. The second half of this course is on transformation groups of manifolds; i.e., groups of diffeomorphism of homeomorphism of manifolds. Those groups share some common properties with Lie groups. We will present basic properties of those groups: for example, simplicity, perfectness, automatic continuity (recent work of Hurtado, Mann and others), and some version of classification of closed subgroups of transformation groups (some even more recent work of Mann and myself).
We will follow two books/papers:
1. John L. Harer: The cohomology of moduli space of curves
2. Augustin Banyaga: The structure of classical diffeomorphism groups