Xiao-Lei Liu

Modular invariants of families of curves.

We will talk about modular invariants of families of curves.  We will give the lower bounds of modular invariants, which are sharp for families of curves of genus 2. The beautiful theory of pseudo-periodic maps is built up by Professor Matsumoto and Professor Montesinos, and we will apply this theory to show the classification of the extremal singular fibers for modular invariants, according to the work by Professor Ashikaga and  Professor Ishizaka.  Moreover,  for the modular invariants corresponding to boundary divisors of the moduli space of curves,   we will discuss their topological meanings.