Applied Mathematics

David Mumford

Professor Emeritus of Applied Mathematics


Ph.D., Harvard University, 1961


I work in the area of machine and natural intelligence, following the statistical approach pioneered by Ulf Grenander, which we call "Pattern Theory". In this approach, thinking is modeled as statistical inference rather than logic and learning results from the accumulation of massive data from interactions with the world. My work concentrates on visual perception which seems to be more approachable than high-level thinking, more complex than auditory and tactile perception yet has been solved by three distinct classes of animals (cephalopods, birds and mammals) so it can't be that hard!

The aspect of this research with which I have been most involved is the construction of probability models for the variables of vision: direct models of the raw images, the shapes of objects, the texture of their surfaces etc. A second question is how to sample and estimate with these models, e.g. compute conditional means and modes. Many approaches involve computing with one or more samples from the distribution which evolve deterministically or stochastically, smoothly or with jumps. Sampling from the distribution can be viewed as feedback, i.e. prior knowledge of high level structures guides the reconstruction of an image. A third set of questions concerns how such statistical estimation may be performed in cortex, in neural nets with feedback. A major open question here is whether the information being handled by neurons is contained only in their firing rates or in the precise timing and synchrony of their spikes in the full network. Another is how cortex handles ambiguity, the need for maintaining multiple hypotheses while waiting for more data.


Faculty Research Profile


1953 Westinghouse Science Talent Search, finalist 

1958-1961 Society of Fellows, Harvard University, Junior Fellow

1974 Fields Medal, International Congress of Mathematics, Vancouver

1975 Elected to the National Academy of Sciences

1978 Honorary Fellow, Tata Institute of Fundamental Research

1983 Honorary Degree of Doctor of Science, University of Warwick

1987-1992 MacArthur Foundation Fellow

1991 Elected Foreign Member, Accademia Nazionale dei Lincei, Rome

1995 Elected Honorary Member, London Mathematical Society

1997 Elected American Philosophical Society

2000 Honorary Degree of DSc., Norwegian University of Science and Technology, Trondheim

2001 Honorary Degree of DSc., Rockefeller University

2004 Elected Foreign Member, Norwegian Academy of Science and Letters

2005 IEEE Longuet-Higgins Prize, CVPR 2005, (with Jayant Shah)

2006 The Shaw Prize in Mathematical Sciences, Hong Kong (shared with Wu Wentsun)

2007 Amer. Math. Society Steele Prize for Mathematical Exposition

2008 Wolf Prize in Mathematics (shared with Pierre Deligne and Phillip Griffiths)

2008 Elected Foreign Member of the Royal Society (UK)

2010 National Medal of Science