Siegel–Veech Measures of Convex Flat Cone Spheres

A classical theorem of Siegel gives the average number of lattice points in bounded subsets of $\mathbb{R}^n$. Motivated by this result, Veech introduced an analogue for translation surfaces, now known as the Siegel–Veech formula. For flat surfaces with irrational cone angles, however, no such formula is available. A convex flat cone sphere is a Riemann sphere equipped with a flat metric with conical singularities, all with cone angles less than $2\pi$. In this talk, I will discuss recent work extending Siegel–Veech theory to this setting and outline the main ideas of the proof.