Mattew Stover

Classifying smooth projective curves by their genus goes back to the
19th century, and by the uniformization theorem (one of the most
fundamental theorems in mathematics) they all admit a Riemannian metric
of constant curvature. A variant I quite like is that they all admit an
orbifold metric of constant curvature -1, and hence are dominated by a
hyperbolic surface. Amazingly the classification of smooth projective
surfaces remains open: the analogues of "genus 0" and "genus 1" are
enumerated, but the "higher genus" box may is a¯\_(ツ)_/¯ with tons
of examples. A question due independently to Farb and Gromov could
actually make the "higher genus" box a primitive recursive enumeration
problem using orbifolds of holomorphic sectional curvature -1,
succinctly generalizing orbifold uniformization of curves. My talk will
be dedicated to setting the stage for their question and using simple
topological arguments (and a giant uniformization hammer) to give some
evidence for it, spinning examples due to Deligne-Mostow and Hirzebruch
(and perhaps my own, if I have time) in this light.