  In this paper we will discuss how one may be able to use mean curvature flow
to tackle some of the central problems in topology in 4-dimensions.
  We will be concerned with smooth closed 4-manifolds that can be smoothly
embedded as a hypersurface in R^5. We begin with explaining why all closed
smooth homotopy spheres can be smoothly embedded. After that we discuss what
happens to such a hypersurface under the mean curvature flow. If the
hypersurface is in general or generic position before the flow starts, then we
explain what singularities can occur under the flow and also why it can be
assumed to be in generic position.
  The mean curvature flow is the negative gradient flow of volume, so any
hypersurface flows through hypersurfaces in the direction of steepest descent
for volume and eventually becomes extinct in finite time. Before it becomes
extinct, topological changes can occur as it goes through singularities. Thus,
in some sense, the topology is encoded in the singularities.

Differential Geometry;Analysis of PDEs;Algebraic Topology;Geometric Topology