Surfaces in 4-dimensional spaces

Topology is the study of spaces of arbitrary dimensions and their properties. The simplest spaces, called manifolds, locally resemble Euclidean spaces, such as the infinite line, or the infinite plane. Curiously, manifolds of dimension at least five behave similarly to one another, while lower dimensional spaces are less controlled. Seminal work of Freedman from 1982 shed light on 4-dimensional manifolds. Unfortunately this is not well-understood by the mathematical community. Recent work provides an accessible exposition of these beautiful ideas as well as extensions and generalizations.