Geometric measure theory
What is it?
Geometric measure theory is concerned with investigating the structure of surfaces from a measure-theoretic viewpoint. Since the notion of a surface (in an appropriately general sense) appears in many different settings in mathematics, it is unsurprising that GMT has applications in many areas of modern mathematics including: PDEs, Harmonic analysis and variational problems.
GMT is traditionally considered to be a hard subject. This is primarily because, although many of the ideas involved are simple, in order to work at the level of generality that we do, a lot of technicalities need to be considered and understood in order to prove useful results.
If you would like to find out a little about my particular areas of interest, then my research page is for you.
To gain a brief overview of the subject, you may like to look at an article that I wrote several years ago for the Kluwer Encyclopedia of Mathematics. It is now available online from Springer.
Kenneth Brakke has developed a very interesting program called Evolver. The program is a sophisticated and useful tool for finding numerical solutions to the minimisation problems which are typically of interest in Geometric measure theory. The program has been successfully used in analysing the behaviour of fuel in cylindrical tanks; and confirming that the stable configuration of two adjacent soap bubbles is the traditional double bubble.
Many books and articles have been written of widely varying levels. Below I give a few books which I have found interesting and helpful.
Geometric measure theory, H Federer, Springer-Verlag 1969
This is the classic text in the subject but is not an easy read. It contains, in great generality, a detailed explanation of the state of the subject in the late sixties.
I cannot recommend this book too highly! It is very hard going but contains beautiful proofs of deep results. Although out of print for some time it has recently been republished by Springer-Verlag in their Classics of mathematics series.
Geometry of sets and measures in Euclidean spaces, P Mattila, CUP 1995
This is a much more modern text than Federer, above, and concentrates on providing a comprehensible introduction to the more recent developments in GMT. The book provides an accessible description of the theory of tangent measures. These were introduced by David Preiss in 1987 and were used by him to solve some of the outstanding conjectures concerning the regularity of measures in Euclidean spaces.
Another book which I heartily recommend!
Lectures on geometric measure theory, L Simon, Australian National University, Centre for Mathematical Analysis 1984
This is a book which outlines the development of the subject since Federer's book was published. It contains many of the results from Federer's book but often gives different proofs.
This book is hard to obtain, which is a pity since it is much more of a direct successor to Federer's book than the one by Mattila is.
Fractal Geometry, K J Falconer, Wiley 1990
As its title suggests, this book concentrates mainly on results applicable to fractals. However, many of the methods it describes were originally developed in the context of GMT and the book is worth looking at. (In fact, there are probably good grounds for claiming much of fractal geometry as a sub-subject of GMT...)
Techniques in Fractal Geometry, K J Falconer, Wiley 1997
This is the sequel to the previous book. It describes how ideas from many areas of mathematics may be used in the study of fractals.
Its main interest for us is the chapter where it introduces tangent measures and gives a clear demonstration of their use.
Plateau's Problem, F J Almgren, Benjamin 1966
This is a delightful little book which provides an intuitive explanation of the theory of varifolds. There are some nice diagrams and clear explanations of the fundamental problems which need solving if we are to derive a satisfactory theory of surfaces. It has recently been republished by the American Mathematical Society.
Geometric measure theory: a beginner's guide, F Morgan, Academic Press 1987
This is an ideal companion book to Federer's tome and acts as a guide to this book, concentrating on explaining the intuitive ideas underlying the subject.