Colloquium

Moduli spaces of smooth complex algebraic curves have been intensively studied since the 19th century from many points of view. Their fundamental groups, being mapping class groups, are among the best understood groups in mathematics. In this talk I will give a historical overview of the geometric study of these groups, and explain how new work with Daniel Allcock (arxiv.org/abs/2609.14416) extends much of this geometric understanding to the fundamental group of another 19th century moduli space, namely the moduli space of smooth complex cubic surfaces. The proof also applies to some other moduli spaces, and gives many new corollaries. Most of the talk will be devoted to motivation and background (some from the "What is..." article https://arxiv.org/abs/2601.15410), and connections to different areas of mathematics will be highlighted.