David Mumford

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David Bryant Mumford (born 11 June 1937) is an American

Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started an experimental school in Tanzania and worked for the then newly created United Nations.^[2]

He attended

Work in algebraic geometry

Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on

Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and a study of Goro Shimura's papers from the 1960s.

Mumford's research did much to revive the classical theory of theta functions, by showing that its algebraic content was large, and enough to support the main parts of the theory by reference to finite analogues of the Heisenberg group. This work on the equations defining abelian varieties appeared in 1966–7. He published some further books of lectures on the theory.

He also was one of the founders of the

In a sequence of four papers published in the American Journal of Mathematics between 1961 and 1975, Mumford explored pathological behavior in algebraic geometry, that is, phenomena that would not arise if the world of algebraic geometry were as well-behaved as one might expect from looking at the simplest examples. These pathologies fall into two types: (a) bad behavior in characteristic p and (b) bad behavior in moduli spaces.

Characteristic-p pathologies

Mumford's philosophy in characteristic p was as follows:

A nonsingular characteristic p variety is analogous to a general non-Kähler complex manifold; in particular, a projective embedding of such a variety is not as strong as a

Kähler metric on a complex manifold, and the Hodge–Lefschetz–Dolbeault theorems on sheaf cohomology

break down in every possible way.

In the second Pathologies paper, Mumford finds that the Hilbert scheme parametrizing space curves of degree 14 and genus 24 has a multiple component. In the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves.

These sorts of pathologies were considered to be fairly scarce when they first appeared. But Ravi Vakil showed in his paper "Murphy's law in algebraic geometry" has shown that Hilbert schemes of nice geometric objects can be arbitrarily "bad", with unlimited numbers of components and with arbitrarily large multiplicities (Invent. Math. 164 (2006), 569–590).

Classification of surfaces

In three papers written between 1969 and 1976 (the last two in collaboration with

Once these adjustments are made, the surfaces are divided into four classes by their Kodaira dimension, as in the complex case. The four classes are: a) Kodaira dimension minus infinity. These are the ruled surfaces. b) Kodaira dimension 0. These are the

Mumford was awarded a

• Westinghouse Science Talent Search
  finalist, 1953.
• Junior Fellow at Harvard from 1958 to 1961.
• Elected to the
  National Academy of Sciences
  in 1975.
• Honorary Fellow from Tata Institute of Fundamental Research in 1978.
• Honorary D. Sc. from the University of Warwick in 1983.
• Foreign Member of Accademia Nazionale dei Lincei, Rome, in 1991.
• Honorary Member of London Mathematical Society in 1995.
• Elected to the American Philosophical Society in 1997.^[10]
• Honorary D. Sc. from Norwegian University of Science and Technology in 2000.^[11]
• Honorary D. Sc. from Rockefeller University in 2001.
• Longuet-Higgins Prize
  in 2005 and 2009.
• Foreign Member of
  The Royal Society
  in 2008.
• Foreign Member of the Norwegian Academy of Science and Letters.^[12]
• Honorary Doctorate from Brown University in 2011.^[13]
• 2012
  BBVA Foundation Frontiers of Knowledge Award
  in the Basic Sciences category (jointly with Ingrid Daubechies).
• Honoris Causa University of Hyderabad, India 2012

He was elected President of the International Mathematical Union in 1995 and served from 1995 to 1999.

See also

Notes

Publications

• Lectures on Curves on Algebraic Surfaces (with George Bergman), Princeton University Press, 1964.
• Geometric Invariant Theory, Springer-Verlag, 1965 – 2nd edition, with J. Fogarty, 1982; 3rd enlarged edition, with F. Kirwan and J. Fogarty, 1994.
• Mumford, David (1999) [1967], The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.), Berlin, New York:
  MR 1748380
• Abelian Varieties, Oxford University Press, 1st edition 1970; 2nd edition 1974.
• Six Appendices to Algebraic Surfaces by Oscar Zariski – 2nd edition, Springer-Verlag, 1971.
• Toroidal Embeddings I (with G. Kempf, F. Knudsen and B. Saint-Donat), Lecture Notes in Mathematics #339, Springer-Verlag 1973.
• Curves and their Jacobians , University of Michigan Press, 1975.
• Smooth Compactification of Locally Symmetric Varieties (with A. Ash, M. Rapoport and Y. Tai, Math. Sci. Press, 1975)
• Algebraic Geometry I: Complex Projective Varieties, Springer-Verlag New York, 1975.
• Tata Lectures on Theta (with C. Musili, M. Nori, P. Norman, E. Previato and M. Stillman), Birkhäuser-Boston, Part I 1982, Part II 1983, Part III 1991.
• Filtering, Segmentation and Depth (with M. Nitzberg and T. Shiota), Lecture Notes in Computer Science #662, 1993.
• Two and Three Dimensional Pattern of the Face (with P. Giblin, G. Gordon, P. Hallinan and A. Yuille), AKPeters, 1999.
• Mumford, David; Series, Caroline; Wright, David (2002), Indra's Pearls: The Vision of Felix Klein,
  MR 1913879 Indra's Pearls: The Vision of Felix Klein
• Selected Papers on the Classification of Varieties and Moduli Spaces, Springer-Verlag, 2004.
• Mumford, David (2010), Selected papers, Volume II. On algebraic geometry, including correspondence with Grothendieck, New York: Springer,
  MR 2741810
• Mumford, David; Desolneux, Agnès (2010), Pattern Theory: The Stochastic Analysis of Real-World Signals, A K Peters/CRC Press,
  MR 2723182
• Mumford, David; Oda, Tadao (2015), Algebraic geometry. II., Texts and Readings in Mathematics, vol. 73, New Delhi: Hindustan Book Agency,
  MR 3443857

External links

Wikimedia Commons has media related to David Mumford.

Wikiquote has quotations related to David Mumford.

• O'Connor, John J.; Robertson, Edmund F., "David Mumford", MacTutor History of Mathematics Archive, University of St Andrews
• David Mumford at the Mathematics Genealogy Project
• Mumford's page at Brown University