Hurwitz theory and the double ramification cycle

TL;DR

This survey explores the relationship between Hurwitz theory and Gromov-Witten theory, focusing on the double ramification cycle and its algebraic, combinatorial, and intersection-theoretic properties, highlighting recent results and open problems.

Contribution

It provides a comprehensive overview of the interactions between Hurwitz theory and the double ramification cycle, including new conjectures and open questions in the field.

Findings

Connections between Hurwitz numbers and intersection theory

Survey of recent results on the double ramification cycle

Identification of open problems and conjectures in the area

Abstract

This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several results by many groups of people on the subject, but, perhaps more importantly, collect a number of conjectures and problems which are still open.