The mapping class groups are natural generalizations of SL(2,Z) to surfaces with higher genera. Remarkably, many properties of SL(2,Z) that seem to depend on linear algebra extend to topological analogs on MCG (for example, we can classify individual element in MCG in the same way we classify hyperbolic isometries). In this talk, I will try to answer the following three questions. (1) Why should we care about MCG? (2) What are the similarity and the difference that occur when we pass from genus 1 to higher genus? (3) How can surfaces bundle, representation theory, and Teichmuller space help us understand MCG, and possibly vice versa? If time permits, I will discuss a representation theoretic version of the RiemannHurwitz formula.
This will be a practice talk for my topic presentation. To express my gratitude for those who come, ice cream will be served.
