Isoperimetric inequalities have been studied since ancient times, originating in the famous "Dido problem" asking which plane figure encloses the greatest area for a fixed perimeter. The answer, a circle, was known to the ancient Greeks, but it was not until the 19th century that Karl Weierstrass gave a rigorous proof sparking the development of the calculus of variations. During the early 20th century, the higher dimensional Euclidean isoperimetric inequality was resolved along with the isoperimetric inequality in constant curvature surfaces and to higher dimensional constant curvature spaces. For variable curvature surfaces, results characterising optimal domains were few and far between until the late 20th century where many results were obtained be a number of authors, and new novel proofs of the classic isoperimetric inequality were also given. In the mean time, during the middle of the 20th century, Geometric Measure Theory (GMT) was developed providing a broad framework and powerful tools for attacking these kinds of problems. Many eminent mathematicians have studied these absorbing problems and such study has lead to the development of a vast array of beautiful mathematics, combining geometry, analysis, topology, and functions that lack full differentiability properties.
The topic of this talk will be on the Isoperimetric Profile developed by several authors in the 1980's and actively studied today. In generalising the above discussion to other spaces such as manifolds and the rectifiable currents studied in GMT, one must deal with the fact that no scaling properties such as in Euclidean space are available. The isoperimetric profile was developed to capture the notion of isoperimetric inequality at all scales. I will describe some background and history of isoperimetric inequalities and introduce the isoperimetric profile. Then I will the relationship between the isoperimetric profile, curvature, convexity and topology and discuss some applications and open problems. One of the most important issues arising is that one encounters spaces of low regularity, and the isoperimetric profile is in general not differentiable. Much theory has been developed to deal with this and I will give a very gentle introduction to some notions from GMT and the theory of convex functions that allow one to show the low regularity is mostly benign. This talk will be accessible to a broad audience of all backgrounds, particularly including students.