Plática especial: "Geometric loci in moduli spaces of curves"

The study of subvarieties of moduli spaces of curves has classically played a crucial role towards the discovery of many important results on the geometry of curves. While the structure of subvarieties of codimension one is known to reflect the birational geometry of the variety, very little is known on the structure of cohomology classes of higher codimension. In this talk I will give an introduction for a general audience to the enumerative geometry of moduli spaces of curves, and then present a selection of my results on higher codimensional subvarieties. I will present explicit formulas for Hurwitz subvarieties of codimension two; discuss the extremality of some effective classes (joint work with Dawei Chen); and present a surprising formula for some extremal classes of high codimension expressed as a sum over graphs indexing boundary strata of moduli spaces of stable curves (joint work with Renzo Cavalieri). These results explore the interplay of geometry and combinatorics of curves and their moduli. I will conclude discussing future directions.