Topology

Michael Hutchings introduces Reeb vector fields and explains why their periodic orbits are substantially better understood in three dimensions through Seiberg-Witten theory. The colloquium develops Irie's closing lemma, which creates a periodic orbit through a selected region after a small perturbation, and discusses a quantitative refinement connecting the perturbation size to an upper bound on the orbit period.

Two-dimensional topological order has caught much attention in connection to topological phases of matter and topological quantum computation, in particular. It is known that tensor networks are useful tools to study mathematical structures appearing in these topics. We present recent understanding of them from a viewpoint of operator algebras. No knowledge on operator algebras are assumed.