The study of isoperimetric inequalities has a long history,
it's humble beginnings in Ancient Greek mathematics belying a deep and
rich theory. A major tool in the study of the isoperimetric profile is
the Calculus of Variations. Variational arguments lead to weak
differential inequalities for the isoperimetric profile, which allows
analytical tools such as the maximum principle to be employed. Of
central importance here is the connection with curvature, which is
intimately connected with the topology of isoperimetric regions. I
will survey some of the results in this direction, paying particular
attention to the interplay of the isoperimetric profile and curvature
flows which is the focus of my current research.