Puncture-forgetting maps play an important role in the study of Teichmüller spaces, mapping class groups, and curves on surfaces. In my talk at KAIST last winter, I introduced the basic idea of puncture-forgetting maps for measured foliations and outlined the proof of the main result. In this talk, to keep the presentation self-contained, we will begin by reviewing the relevant definitions and the main theorems. We will then discuss recent developments in this theory and present a new application to the universal curve over Teichmüller space, viewed as the (infinite-volume) quotient of Teichmüller space by the point-pushing mapping class group. Then, we will also discuss several related questions and directions for future research. This is joint work with Jeremy Kahn.

Wall's stabilization principle suggests that exotic phenomena in dimension four in the orientable category disappear after taking connected sums with sufficiently many S2xS2. Since most known exotic pairs of closed 4-manifolds become diffeomorphic after one stabilization, a natural question was: is a single S2xS2 enough? Recently, Jianfeng Lin constructed an exotic diffeomorphism on a closed 4-manifold-a diffeomorphism topologically isotopic to the identity but not smoothly isotopic-that survives one stabilization. In this talk, we provide a relative exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations. This gives the first exotic phenomenon in the orientable category that survives two stabilizations. The obstruction to stabilization comes from equivariant Seiberg–Witten theory, together with a version of lattice homology. I will also survey some background and recent developments in equivariant gauge theory. This is joint work with Sungkyung Kang and JungHwan Park.

Trees generalize in (at least) three different ways, CAT(0) cube complexes which is a fine metric notion, hyperbolic spaces which is a coarse metric notion and non-Hausdorff trees which is a topological notion that arises naturally when studying Anosov flows on closed three manifolds. I will discuss analogies between the three contexts with focus on recent joint work with Barthelm’e, Mann and Paulet where we build a counterpart of Hagen’s contact graph for bifoliated planes and use it to derive several genericity results for groups acting on bifoliated planes by foliation-preserving homeomorphisms.