# Symplectic four-manifolds through branched coverings

May 14 to May 18, 2018

at the

American Institute of Mathematics, San Jose, California

organized by

Tye Lidman, Daniel Ruberman, and Laura Starkston

## Original Announcement

This workshop will be devoted to studying problems in the topology of symplectic four-manifolds by using connections with gauge theory, holomorphic curves, and algebraic geometry. It is known that every symplectic four-manifold arises from a branched cover over a symplectic surface in $\mathbb{C}P^2$, and thus both branched covering constructions and symplectic curves in $\mathbb{C}P^2$ provide hands-on techniques to understand many important problems in symplectic 4-manifold topology. A significant aim of the workshop is to import and extend techniques from the algebro-geometric study of complex curves in complex surfaces and their birational classifications into the category of symplectic four-manifolds. A similar goal applies to the use of gauge theory/Floer homology invariants for similar problems, as recently these invariants have made new progress towards understanding curves in $\mathbb{C}P^2$ and the algebraic topology of symplectic 4-manifolds. The focus will be on concretely building connections between these areas and symplectic topology while working directly on specific problems in symplectic topology posed throughout the workshop.

The main topics for the workshop are:

1. Understanding symplectic surfaces in $\mathbb{C}P^2$, especially which types of singularities can occur and the number of isotopy classes of singular symplectic surfaces

2. Understanding the extent to which the Bogomolov-Miyaoka-Yau inequality from complex algebraic geometry extends to symplectic four-manifolds
3. Determining when a smooth four-manifold admits a branched cover which is symplectic and in connection, exploring how gauge theoretic invariants of four-manifolds behave under branched coverings

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.