David Mumford

Pattern Theory started in the 70’s with the ideas of Ulf Grenander and his school at Brown. The aim is to analyze from a statistical point of view the patterns in all ‘signals' generated by the world, whether they be images, sounds, written text, DNA or protein strings, spike trains in neurons, time series of prices or weather, etc. Pattern theory proposes that the types of patterns – and the hidden variables needed to describe these patterns – found in one class of signals will often be found in the others and that their characteristic variability will be similar. The underlying idea is to find classes of stochastic models which can capture all the patterns that we see in nature, so that random samples from these models have the same ‘look and feel' as the samples from the world itself. Then the detection of patterns in noisy and ambiguous samples can be achieved by the use of Bayes's rule, a method that can be described as ‘analysis by synthesis'.

My book with Agnes Desolneux, entitled “Pattern Theory: the Stochastic Analysis of Real Woirld Signals”, is expected to appear this year, published by AKPeters.

Current research

A) Object recognition requires that you know when two shapes are ‘similar’. But what does similar mean? The mathematician says: make the set of all (two dimensional, three dimensional or higher) shapes into the points of an infinite-dimensional space and put a metric on this space. If shape means an open subset of Euclidean space with smooth boundary, then this space should be a Banach manifold, but a highly non-linear one. What structure does this space have? I am looking at various metrics and the associated completions of the space of shapes; the geodesics in these metrics and the curvature of the space; examples and applications to object recognition.

B) The History of Mathematics is usually taught from a very Western-centric point of view. Without a doubt the Greek contributions were huge, but once you open your mind to the fact that mathematical truth can and has been discovered by many routes other than Euclid-style proofs (think of calculus in the 17^th-18^th century), you find a much richer picture in which Mesopotamia, India and China all developed deep mathematical ideas, sometimes before, sometimes after similar ideas appeared in the West. I have been studying Indian math in particular, which is astonishing in both its similarities and differences from the West (see my review of Plofker’s recent History Mathematics in India).

Indra’s Pearls, with Caroline Series and Dave Wright, Cambridge Univ. Press

3d-lattice2v2 From the Introduction:

This is a book about serious mathematics, but one, which we have written primarily for non-mathematicians. It is an account of our exploration of a family of symmetrical but infinitely convoluted sets, part of the modern investigation of how chaos evolves from very simple rules, producing intricate complexity on every scale from the very large to the very small. In our case, two rules, each on its own producing a pair of spirals, are allowed to interact. Our scheme is not at all arbitrary; it forms parts of a century old mathematical dream, involving much of the deep mathematics of the 19th century, conceived by the great German geometer Felix Klein. With the aid of modern computers for rendering the results, the answers to our `What if...?' questions turned to be not only intellectually fascinating but also strikingly beautiful. Sometimes the outcome is simple, sometimes it is total disorder and sometimes -- and this is the most exciting case -- it has layer upon layer of structure teetering on the very brink of chaos. There is no religion in our book but we were amazed at how well our constructions reflected the ancient Buddhist metaphor of Indra's net. Mathematicians often use the word `beautiful' in talking about their proofs and ideas, but in this case our judgment has been confirmed by a number of unbiased and definitely non-mathematical people.

Most mathematics is accessible, as it were, only by crawling through a long tunnel in which you laboriously build up your vocabulary and skills as you abstract your understanding of the world. But the mathematics behind the figures we drew turned out not to need too much in the way of preliminaries. So long as you got through high school algebra with some confidence, everything we say should be understandable, given a bit of careful reading here and there. And if not, then browsing through the figures alone should give a sense of our journey. Our dream is that this book will reveal to a larger audience that mathematics is not alien, cold and remote but just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty.

8 See Indra’s home page (stills) with movies here.