How Riemann surfaces lead to algebra (FARS)

Inspired by physicist, Atiyah and Segal initiated an area of mathematics which is the sometimes called "topological field theory." While this can be a useful way of thinking about algebra using pictures, it is essentially impossible to produce genuine new examples using only topology. The additional to that is required is analysis, in the form of Gromov's and Floer's theory of pseudo-holomorphic curves. I will illustrate with examples.