Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera

TL;DR

This paper discusses conjectures about the asymptotic behavior of volumes and Siegel-Veech constants of moduli spaces of Abelian differentials as the genus increases, supported by numerical evidence and recent progress.

Contribution

It formulates conjectures on large genus asymptotics and reviews recent advances and numerical evidence related to these conjectures.

Findings

Numerical evidence supports the conjectured asymptotic behavior.

Recent progress has been made towards proving the conjectures.

The paper summarizes the current state of research in this area.

Abstract

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.