Geometry and analysis of moduli spaces of Riemann surfaces

The moduli space of compact Riemann surfaces of genus 1 is equal to the basic locally symmetric space $SL(2, Z) \ H^2$, the quotient of the upper halfplane $H^2$ by the modular group $SL(2, Z)$. Therefore, the moduli spaces $M_g$ of compact Riemann surfaces of genus $g$ share the common root with locally symmetric spaces and other similarity. In this talk I will discuss some results on the geometry, topology and analysis of the moduli spaces partially motivated by this analogy.