When is topology the same as algebra? In my talk I will describe how closed surfaces living in 3-dimensional space can be described using an algebraic structure called a Frobenius algebra. We introduce a little category theory to formalize the process of translating topology into algebra. In particular, the process described above is a functor called a 2-dimensional topological quantum field theory because it axiomatizes various properties that are present in the path integral approach to quantum field theory.
The talk should be accessible to everybody (elementary algebra required).