Knotty algebraic curves

Kyle Hayden (Columbia)

Four-manifold topology is famous for "exotic" phenomena, like non-standard smooth structures on Euclidean 4-space. However, this wild flexibility often evaporates in the presence of geometry, such as complex or symplectic structures. In this talk, we'll explore this phase transition using braids and mapping class groups. As an application, we will produce slice disks in the 4-ball that are isotopic through ambient homeomorphisms but not ambient diffeomorphisms; in fact, these give rise to the first examples of "exotically knotted" algebraic curves. There will be many cartoons.