Combinatorics and Optimization Professor talk - Eric Katz

The genus formula and tropical geometry

Abstract:

A complex homogeneous polynomial in three variables describes an algebraic curve in projective space. This algebraic curve is a topologically a surface. We can ask what is the genus of the surface is in terms of the degree of the polynomial. This is the celebrated genus formula. We will prove it by decomposing the surface into simpler surfaces called pairs-of-pants. In doing so, we will introduce techniques of tropical geometry.