Symplectic four-manifolds through branched coverings

Symplectic four-manifolds through branched coverings

This workshop, sponsored by AIM and the NSF, 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

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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