Pure Maths Colloquium: Renzo Cavalieri

A pragmatic view of tropical geometry regards it as a combinatorialization of complex or algebraic geometric objects such that a surprisingly large number of algebraic invariants may be recovered by the corresponding combinatorial ones. I will survey this connection in the context of the moduli spaces of curves. In recent years the relationship between the algebraic and tropical moduli spaces of curves has been clarified by gaining a concrete understanding of how tropical curves capture the infinitesimal geometry of deformations of algebraic curves. My goal is to give a broad strokes description of the evolution of this story, suitable for a general audience. Time permitting, I will focus on some recent results (joint with Hannah Markwig and Andreas Gross) on the intersection theory on tropical moduli spaces of curves of higher genus.