Georgia Institute of Technology College of Sciences

There exist many different diagrammatic descriptions of 4-manifolds, with the usual claim that "such and such a diagram uniquely determines a smooth 4-manifold up to diffeomorphism". This raises higher order questions: Up to what diffeomorphism? If the same diagram is used to produce two different 4-manifolds, is there a diffeomorphism between them uniquely determined up to isotopy? Are such isotopies uniquely determined up to isotopies of isotopies? Such questions become important if one hopes to use "diagrams" to study spaces of diffeomorphisms between manifolds. One way to achieve these higher order versions of uniqueness is to ask that a diagram uniquely determine a contractible space of 4-manifolds (i.e. a 4-manifold bundle over a contractible space). I will explain why some standard types of diagrams do not do this and give at least one type of diagram that does do this.

An outstanding problem for surface bundles over surfaces, closely related to the symplectic geography problem in dimension four, is to determine for which fiber and base genera there are examples with non-zero signatures. I will report on our recent progress (joint with M. Korkmaz), which resolves the problem for all fiber and base genera except for 18 pairs at the time of writing.

Given an immersed, Maslov-0, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-0, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-0, exact Lagrangian filling with genus g ≥ 1 and p double points can be obtained from such a Lagrangian surgery on a filling of genus g − 1 with p+1 double points. To show this, we establish the connection between the existence of an immersed, Maslov-0, exact Lagrangian filling of a Legendrian Λ that has p double points with action 0 and the existence of an embedded, Maslov-0, exact Lagrangian cobordism from p copies of a Hopf link to Λ. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-0, exact Lagrangian cobordisms between Legendrian links. Joint work with Noemie Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.