Branched covers of the sphere and the moduli space of curves: Geometry, physics, representation theory, combinatorics

The moduli space of curves has been a central tool and object of study in algebraic geometry for many decades, and yet its topology (and intersection theory) has proved remarkably difficult to understand, despite the belief that much of it is in some sense combinatorial. Recent developments in Gromov-Witten theory have established a connection between intersection theory on the moduli space of curves and branched covers of the sphere, bringing to bear ideas from the fields of mathematics mentioned in the title.

I'll describe the moduli space of pointed curves, Witten's conjecture (Kontsevich's theorem), Faber's conjecture, and the Virasoro conjecture, and discuss some of the developments in the area (both old and new), and future prospects. Despite the subject matter, most of the talk will be accessible to a wide audience.