David Bryant Mumford

Born: 11 June 1937 in Worth, Sussex, England

David Mumford's father, William Mumford, was English and [2]:-

... a visionary with an international perspective, who started an experimental school in Tasmania based on the idea of appropriate technology...

Mumford's father worked for the United Nations from its foundations in 1945 and this was his job while Mumford was growing up. Mumford's mother was American and the family lived on Long Island Sound in the United States, a semi-enclosed arm of the North Atlantic Ocean with the New York- Connecticut shore on the north and Long Island to the south.

After attending Exeter School, Mumford entered Harvard University. It was at Harvard that Mumford first became interested in algebraic varieties. He relates in [2]:-

... a classmate said "Come with me to hear Professor Zariski's first lecture, even though we won't understand a word" and Oscar Zariski bewitched me. When he spoke the words "algebraic variety", there was a certain resonance in his voice that said distinctly that he was looking into a secret garden. I immediately wanted to be able to do this too. It led me to 25 years of struggling to make this world tangible and visible.

After graduating from Harvard, Mumford was appointed to the staff there. He was appointed professor of mathematics in 1967 and, ten years later, he became Higgins Professor. He was chairman of the Mathematics Department at Harvard from 1981 to 1984 and MacArthur Fellow from 1987 to 1992.

Mumford's greatest honour was being awarded a Fields Medal at the International Congress in Vancouver in 1974. Tate describes the work that Mumford was awarded the Fields Medal for in [6]. He writes:-

Mumford's major work has been a tremendously successful multi-pronged attack on problems of the existence and structure of varieties of moduli, that is, varieties whose points parameterise isomorphism classes of some type of geometric object. Besides this he has made several important contributions to the theory of algebraic surfaces. ... Mumford has carried forward, after Zariski, the project of making algebraic and rigorous the work of the Italian school on algebraic surfaces. He has done much to extend Enriques' theory of classification to characteristic p > 0, where many new difficulties appear.

Tate goes on to explain in more technical terms Mumford's work in [6] and it is also described in [4]. In the 1980s however, the direction of Mumford's work changed dramatically. He writes in [1] that, after his first wife Erika died:-

... I turned from algebraic geometry to an old love - is there a mathematical approach to understanding thought and the brain? This is applied mathematics and I have to say that I don't think theorems are very important here. I met remarkable people who showed me the crucial role played by statistics, Grenander, Geman and Diaconis.

The article [3] is a survey of this new area that Mumford has worked on. It makes fascinating reading and we quote here some comments from the introduction which give an idea of the scope of the ideas covered:-

The term "Pattern Theory" was introduced by Ulf Grenander in the 70's as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence. ... The problem that "Pattern Theory" aims to solve ... may be described as follows "the analysis of the patterns generated by the world in any modularity, with all their naturally occurring complexity and ambiguity, with the goal of reconstructing the processes, objects and events that produced them and of predicting these patterns when they reoccur".

Mumford has received many honours in addition to the Fields Medal. He received an honorary D.Sc. from the University of Warwick in 1983, an honorary D.Sc. from the Norwegian University of Science and Technology in 2000, and an honorary D.Sc. from Rockefeller University in 2001. He was elected to the National Academy of Sciences in 1975, elected an Honorary Fellow of the Tata Institute of Fundamental Research in 1978, elected a Foreign Member of Accademia Nazionale dei Lincei, Rome, in 1991, elected an Honorary Member of London Mathematical Society in 1995, elected to the American Philosophical Society in 1997, and elected a Foreign Member of the Royal Society in 2008. He was awarded the Shaw Prize in 2006, the Steele Prize for Mathematical Exposition by the American Mathematical Society in 2007, and the Wolf Prize in 2008. He was elected President of the International Mathematical Union in 1995, a position he held until 1999.

Let us mention the book Indra's Pearls: The Vision of Felix Klein which he published with Caroline Series and David Wright in 2002. Vasily Chernecky begins a review of this remarkable book as follows:-

Felix Klein, one of the great nineteenth-century geometers, discovered in mathematics an idea prefigured in Buddhist mythology: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour, so that the whole Universe was mirrored in each pearl. Klein studied infinitely repeated reflections and was led to forms with multiple co-existing symmetries, each simple in itself, but whose interactions produce fractals on the edge of chaos. For a century these images, which were practically impossible to draw by hand, barely existed outside the imagination of mathematicians. However in the 1980s the authors embarked on the first computer exploration of Klein's vision, and in so doing found further extraordinary images of their own. The book is written as a guide to actually coding the algorithms which are used to generate the delicate fractal filigrees, most of which have never appeared in print before.

Finally, to get a feeling for his current interests we mention a lecture he delivered on 11 February 2008. The lecture was entitled What's an infinite dimensional manifold and how can it be useful in hospitals? and Mumford gave the following abstract:-

Morphing faces has become a popular game but what is the math behind it? One way to view it is as the construction of geodesics on an infinite dimensional manifold of shapes. I will try to explain what this means, using simple examples and then go on show why it is proposed as a new tool in the diagnosis of medical conditions.

Article by: J J O'Connor and E F Robertson

September 2009

MacTutor History of Mathematics