David Mumford
David Mumford
Professor Emeritus of Applied Mathematics
Overview
See my home page www.dam.brown.edu/people/mumford.
Brown Affiliations
Research Areas
auditory and tactile perception | Banach manifold | cortical computation | Euclidean space | geodesics | hierarchical Bayesian inference | Indra's Pearls | machine and natural intelligence | modeling | object recognition | pattern theory | probability models | vision | visual cortex | visual perception
Publications
| Helgason, Sigurdur, Scott, Dana, Peters, Alice, Mumford, David. "Remembering Klaus Peters." Notices Amer. Math. Soc., vol. 62, no. 03, 2015, pp. 264-268. |
| BRUVERIS, MARTINS, MICHOR, PETER W., MUMFORD, DAVID. "GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES." Forum of Mathematics, Sigma, vol. 2, 2014. |
| Mulay, Shashikant, Sathaye, Avinash, Abhyankar, Yvonne, Bajaj, Chandrajit, Cutkosky, Steven Dale, Harbater, David, Heinzer, William, Shannon, David, Mumford, David, Thakur, Dinesh. "Shreeram Abhyankar (July 22, 1930–November 2, 2012)." Notices Amer. Math. Soc., vol. 61, no. 10, 2014, pp. 1196. |
| Michor, Peter W., Mumford, David. "A zoo of diffeomorphism groups on $$mathbb{R }^{n}$$ R n." Ann Glob Anal Geom, vol. 44, no. 4, 2013, pp. 529-540. |
| Michor, Peter W., Mumford, David. "On Euler's equation and 'EPDiff'." JGM, vol. 5, no. 3, 2013, pp. 319-344. |
| Tu, Loring W., Gurdian, Rodolfo, Smale, Stephen, Mumford, David, Jaffe, Arthur, Yau, Shing-Tung. "Remembering Raoul Bott (1923–2005)." Notices Amer. Math. Soc., vol. 60, no. 04, 2013, pp. 398. |
| Micheli, Mario, Michor, Peter W, Mumford, David. "Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds." Izv. Math., vol. 77, no. 3, 2013, pp. 541-570. |
| Barnsley, Michael F., Berry, Michael, Frame, Michael, Stewart, Ian, Mumford, David, Falconer, Kenneth, Eglash, Ron, Brothers, Harlan, Lesmoir-Gordon, Nigel, Barrallo, Javier. "Glimpses of Benoît Mandelbrot (1924--2010)." Notices Amer. Math. Soc., vol. 59, no. 08, 2012, pp. 1056. |
| Micheli, Mario, Michor, Peter W., Mumford, David. "Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks." SIAM J. Imaging Sci., vol. 5, no. 1, 2012, pp. 394-433. |
| Barnsley, Michael F., Frame, Michael, Howe, Roger, Stewart, Ian, Mumford, David, Furstenberg, Hillel, Falconer, Kenneth, West, Bruce J., Coppens, Marc-Olivier, Cohen, Nathan, Jaffard, Stéphane, Berry, Michael. "The Influence of Benoît B. Mandelbrot on Mathematics." Notices Amer. Math. Soc., vol. 59, no. 09, 2012, pp. 1208. |
| Akin, Alexander, Mumford, David. "Yu laid out the lands:” georeferencing the Chinese Yujitu [Map of the Tracks of Yu] of 1136." Cartography and Geographic Information Science, vol. 39, no. 3, 2012, pp. 154-169. |
| Mumford, D. "Is there a general structure for grammars?." 2009 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2009. |
| Younes, Laurent, Michor, Peter, Shah, Jayant, Mumford, David. "A metric on shape space with explicit geodesics." Rendiconti Lincei - Matematica e Applicazioni, 2008, pp. 25-57. |
| Feiszli, Matt, Mumford, David. "Shape representation via conformal mapping." Computational Imaging V, 2007. |
| Michor, Peter W., Mumford, David. "An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach." Applied and Computational Harmonic Analysis, vol. 23, no. 1, 2007, pp. 74-113. |
| Zhu, Song-Chun, Mumford, David. "A Stochastic Grammar of Images." Foundations and Trends® in Computer Graphics and Vision, vol. 2, no. 4, 2006, pp. 259-362. |
| Mumford, D. "Mathematics Belongs in a Liberal Education." Arts and Humanities in Higher Education, vol. 5, no. 1, 2006, pp. 21-32. |
| Michor, Peter, Mumford, David. "Riemannian geometries on spaces of plane curves." Journal of the European Mathematical Society, 2006, pp. 1-48. |
| Keen, Linda. "Indra’s pearls. The vision of Felix Klein." The Mathematical Intelligencer, vol. 27, no. 4, 2005, pp. 58-60. |
Research
Research Overview
Please visit my webpages at http://www.dam.brown.edu/people/mumford/index.html
Research Statement
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.
Funded Research
You may contact an Administrative Assistant in the Division of Applied Mathematics for this information.
Background
Education and Training
| Year | Degree | Institution |
|---|---|---|
| 1961 | PhD | Harvard University |
| 1957 | BA | Harvard University |
Honors and Awards
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 Univ. 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
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 Univ. 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
Affiliations
Affiliations
Please see my curriculum vita.
Teaching
Teaching Overview
See my home page www.dam.brown.edu/people/mumford.