4-manifolds (sympletic or not)
Ana Cannas da Silva (Instituto Superior Tecnico, Lisboa)
Course description
We will review the state of the art and open questions on (smooth) 4-manifolds. We will explain the existence on 4-manifolds of structures related to symplectic forms.
Tentative plan:
1st lecture - 4-Manifolds
Whereas (closed simply connected) topological 4-manifolds are completely classified, the panoramam for smooth 4-manifolds is quite wild: the existence of a smooth structure imposes strong topological constraints, yet for the same topology there can be infinite different smooth structures. We will discuss constructions of (smooth) 4-manifolds and invariants to distinguish them.
2nd lecture - Symplectic Geography
Symplectic geography is concerned with existence and uniqueness of symplectic forms on a given manifold. These questions are particularly relevant to 4-dimensional topology and to mathematical physics, where symplectic manifolds occur as building blocks or as key examples. We will describe some constructions of symplectic 4-manifolds and invariants to distinguish them.
3rd lecture - Folded Symplectic Forms
Any orientable 4-manifold admits a folded symplectic form, that is, a closed 2-form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic form by a folding map. We will explain how, for orientable even-dimensional manifolds, the existence of a stable almost complex structure is necessary and sufficient to warrant the existence of a folded symplectic form.