학과 세미나 및 콜로퀴엄
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.
Asymptotically Locally Flat (ALF) Ricci-flat metrics are expected to model certain long-time singularities in four-dimensional Ricci flow, so understanding their stability is essential. In this talk, I will discuss that conformally Kähler, non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. Our work establishes three key tools in this setting: a Fredholm theory for the Laplacian on ALF metrics, the preservation of the ALF structure along the Ricci flow, and an extension of Perelman’s λ-functional to ALF metrics. This is joint work with Tristan Ozuch.
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.