Symplectic structures roughly lie between smooth topology and complex geometry and exhibit both flexible and rigid phenomena. However, we still do not have a good understanding of what a general symplectic manifold should look like.

In this talk, I will begin by surveying some classical results of Gromov and McDuff from the 1980s, which use pseudoholomorphic curves to classify symplectic manifolds containing certain distinguished submanifolds.

I will then discuss a question of Donaldson in the 1990s, which relates smooth structures in dimension four to symplectic structures in dimension six.

Finally, I will describe some recent refinements of the Gromov-McDuff techniques and their applications to classification problems in the compact setting, which has many interactions with low-dimensional topology.

Spaces of complex polynomials contain a surprising amount of combinatorial structure, particularly in connections to noncrossing partitions. In this talk we will discuss how to extract this noncrossing data from a polynomial and how it determines a compactification and cell structure for a space of polynomials. This is joint work with Jon McCammond.