The subfield of low-dimensional topology colloquially called "3.5-dimensional topology" studies closed 3-manifolds through the eyes of the 4-manifolds that they bound. This talk focusses on Casson's question of which rational homology 3-spheres bound rational homology 4-balls. Since rational homology 3-spheres bounding rational homology 4-balls are a rare phenomenon, we will discuss how to construct examples.

TBA

TBA

We see how smooth versions of peg problems give rise to constructions in symplectic topology. (No knowledge of symplectic topology is required).

We have a leisurely discussion of the Toeplitz Square peg problem and a little of its history.

Puncture–forgetting maps have been studied for a variety of objects, including Teichmüller spaces, mapping class groups, and closed curves. In this talk, we discuss several ideas of forgetting punctures in measured foliations, which give rise to upper semi-continuous maps between spaces of measured foliations. In the proof, we introduce complexes of pre-homotopic multicurves and show that they are hyperbolic CAT(0) cube complexes. We then study the action of point-pushing mapping classes on these complexes. This theory is motivated by applications to Teichmüller geodesics and the dynamics of post-critically finite rational maps. This is joint work with Jeremy Kahn.

A fundamental problem in low-dimensional topology is to find the minimal genus of embedded surfaces in a 3-manifold or 4-manifold, in a given homology class. Ni and Wu solved a 3-dimensional minimal genus problem for rationally null-homologous knots. In this talk, we will discuss an analogous 4-dimensional minimal genus problem for rationally null-homologous knots. This is a joint work with Zhongtao Wu.