Let's Get Tropical! (with Moduli Spaces of Curves)

Our own Andy Fry will talk about tropical mathematics. Tropical mathematics replaces addition (a+b) with taking the minimum (min{a,b}) and multiplication (ab) with addition (a+b). When we do this, lines and curves transform (tropicalize) into piecewise-linear objects. A strength of tropical geometry is that it allows us to look at a "linear" skeleton of a potentially complicated geometric object, reducing algebro-geometric questions to those of combinatorics. A strong trend in modern algebraic geometry is the study of moduli (parameter) spaces. Broadly, a moduli space parameterizes geometric objects, and we can define algebraic moduli spaces and tropical moduli spaces independently. My research investigates tropicalization questions involving moduli spaces of curves, that is, which algebraic moduli spaces "tropicalize" to their tropical counterparts. In this talk, I will introduce tropical mathematics, including tropical arithmetic, algebra, and geometry. Then I will define the tropical moduli space of curves and briefly describe my own research. This talk will be accessible to those taking Calculus 3 or Linear Algebra.