David Bryant Mumford (born 11 June 1937) is an American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
known for his work in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and then for research into
vision and
pattern theory. He won the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
and was a
MacArthur Fellow. In 2010 he was awarded the
National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at
Brown University
Brown University is a private research university in Providence, Rhode Island. Brown is the seventh-oldest institution of higher education in the United States, founded in 1764 as the College in the English Colony of Rhode Island and Providenc ...
.
Early life
Mumford was born in
Worth, West Sussex in
England
England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe ...
, of an English father and American mother. His father William started an experimental school in
Tanzania
Tanzania (; ), officially the United Republic of Tanzania ( sw, Jamhuri ya Muungano wa Tanzania), is a country in East Africa within the African Great Lakes region. It borders Uganda to the north; Kenya to the northeast; Comoro Islands ...
and worked for the then newly created
United Nations
The United Nations (UN) is an intergovernmental organization whose stated purposes are to maintain international peace and security, develop friendly relations among nations, achieve international cooperation, and be a centre for harmoni ...
.
He attended
Phillips Exeter Academy
(not for oneself) la, Finis Origine Pendet (The End Depends Upon the Beginning) gr, Χάριτι Θεοῦ (By the Grace of God)
, location = 20 Main Street
, city = Exeter, New Hampshire
, zipcode ...
, where he received a
Westinghouse Science Talent Search
Westinghouse may refer to:
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*Westinghouse Electric Corporation, the company that manages the Westinghouse brand, with licensees:
** Westinghouse Electric Company, providing nuclear power-related services
**Westingho ...
prize for his relay-based computer project. Mumford then went to
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...
, where he became a student of
Oscar Zariski. At Harvard, he became a
Putnam Fellow in 1955 and 1956.
He completed his
PhD PHD or PhD may refer to:
* Doctor of Philosophy (PhD), an academic qualification
Entertainment
* '' PhD: Phantasy Degree'', a Korean comic series
* '' Piled Higher and Deeper'', a web comic
* Ph.D. (band), a 1980s British group
** Ph.D. (Ph.D. al ...
in 1961, with a thesis entitled ''Existence of the moduli scheme for curves of any genus''. He married Erika, an author and poet, in 1959 and they had four children, Stephen, Peter, Jeremy, and Suchitra. He currently has seven grandchildren.
Work in algebraic geometry
Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s, with a theory summed up in his book ''
Geometric Invariant Theory'', on the equations defining an
abelian variety, and on
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s.
His books ''Abelian Varieties'' (with
C. P. Ramanujam
Chakravarthi Padmanabhan Ramanujam (9 January 1938 – 27 October 1974) was an Indian mathematician who worked in the fields of number theory and algebraic geometry. He was elected a fellow of the Indian Academy of Sciences in 1973.
Like his n ...
) and ''Curves on an Algebraic Surface'' combined the old and new theories. His lecture notes on
scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise
Éléments de géométrie algébrique, the only accessible introduction. They are now available as ''The Red Book of Varieties and Schemes'' ().
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 In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension ''d'' ...
appeared in 1966–7. He published some further books of lectures on the theory.
He also was one of the founders of the
toroidal embedding theory; and sought to apply the theory to
Gröbner basis techniques, through students who worked in algebraic computation.
Work on pathologies in algebraic geometry
In a sequence of four papers published in the ''
American Journal of Mathematics'' between 1961 and 1975, Mumford explored pathological behavior in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, 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 first Pathologies paper, Mumford finds an everywhere regular differential form on a smooth projective surface that is not closed, and shows that Hodge symmetry fails for classical
Enriques surfaces in characteristic two. This second example is developed further in Mumford's third paper on classification of surfaces in characteristic ''p'' (written in collaboration with
E. Bombieri
Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathem ...
). This pathology can now be explained in terms of the
Picard scheme
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a glob ...
of the surface, and in particular, its failure to be a
reduced scheme, which is a theme developed in Mumford's book "Lectures on Curves on an Algebraic Surface". Worse pathologies related to p-torsion in
crystalline cohomology were explored by
Luc Illusie
Luc Illusie (; born 1940) is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic ...
(Ann. Sci. Ec. Norm. Sup. (4) 12 (1979), 501–661).
In the second Pathologies paper, Mumford gives a simple example of a surface in characteristic ''p'' where the
geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...
is non-zero, but the second Betti number is equal to the rank of the
Néron–Severi group In algebraic geometry, the Néron–Severi group of a variety is
the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is nam ...
. Further such examples arise in
Zariski surface theory. He also conjectures that the
Kodaira vanishing theorem is false for surfaces in characteristic ''p''. In the third paper, he gives an example of a
normal surface for which Kodaira vanishing fails. The first example of a smooth surface for which Kodaira vanishing fails was given by
Michel Raynaud in 1978.
Pathologies of moduli spaces
In the second Pathologies paper, Mumford finds that the
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a ...
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 recently,
Ravi Vakil
Ravi D. Vakil (born February 22, 1970) is a Canadian-American mathematician working in algebraic geometry.
Education and career
Vakil attended high school at Martingrove Collegiate Institute in Etobicoke, Ontario, where he won several mathemati ...
in a paper called "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
Enrico Bombieri), Mumford extended the
Enriques–Kodaira classification of smooth
projective surfaces from the case of the complex
ground field to the case of an
algebraically closed ground field of characteristic ''p''. The final answer turns out to be essentially the same as the answer in the complex case (though the methods employed are sometimes quite different), once two important adjustments are made. The first is that one may get "non-classical" surfaces, which come about when ''p''-torsion in the
Picard scheme
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a glob ...
degenerates to a non-reduced group scheme. The second is the possibility of obtaining
quasi-elliptic surfaces in characteristics two and three. These are surfaces fibred over a curve where the general fibre is a curve of
arithmetic genus one with a cusp.
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
K3 surfaces,
abelian surface In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bili ...
s, hyperelliptic and
quasi-hyperelliptic surfaces, and
Enriques surfaces. There are classical and non-classical examples in the last two Kodaira dimension zero cases.
c) Kodaira dimension 1. These are the elliptic and
quasi-elliptic surfaces not contained in the last two groups.
d) Kodaira dimension 2. These are the
surfaces of general type.
Awards and honors
Mumford was awarded a
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
in 1974. He was a
MacArthur Fellow from 1987 to 1992. He won the
Shaw Prize in 2006. In 2007 he was awarded the
Steele Prize
The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories.
The prizes have ...
for Mathematical Exposition by the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
. In 2008 he was awarded the
Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of nati ...
; on receiving the prize in Jerusalem from
Shimon Peres, Mumford announced that he was donating half of the prize money to
Birzeit University
Birzeit University (BZU; ar, جامعة بيرزيت) is a public university in the West Bank, in the State of Palestine, registered by the Palestinian Ministry of Social Affairs as charitable organization. It is accredited by the Ministry of ...
in the
Palestinian territories
The Palestinian territories are the two regions of the former British Mandate for Palestine that have been militarily occupied by Israel since the Six-Day War of 1967, namely: the West Bank (including East Jerusalem) and the Gaza Strip. The ...
and half t
Gisha an Israeli organization that promotes the right to freedom of movement of Palestinians in the Gaza Strip.
He also served on the Mathematical Sciences jury for the
Infosys Prize in 2009 and 2010. In 2010 he was awarded the
National Medal of Science. In 2012 he became a fellow of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
.
There is a long list of awards and honors besides the above, including
*
Westinghouse Science Talent Search
Westinghouse may refer to:
Businesses Current companies
*Westinghouse Electric Corporation, the company that manages the Westinghouse brand, with licensees:
** Westinghouse Electric Company, providing nuclear power-related services
**Westingho ...
finalist, 1953.
*
Junior Fellow at Harvard from 1958 to 1961.
*Elected to the
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...
in 1975.
*Honorary Fellow from
Tata Institute of Fundamental Research in 1978.
*Honorary D. Sc. from the
University of Warwick
, mottoeng = Mind moves matter
, established =
, type = Public research university
, endowment = £7.0 million (2021)
, budget = £698.2 million (2020 ...
in 1983.
*Foreign Member of
Accademia Nazionale dei Lincei
The Accademia dei Lincei (; literally the "Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rom ...
,
Rome
, established_title = Founded
, established_date = 753 BC
, founder = King Romulus ( legendary)
, image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg
, map_caption ...
, in 1991.
*Honorary Member of
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...
in 1995.
*Elected to the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
in 1997.
*Honorary D. Sc. from
Norwegian University of Science and Technology
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
* Demographics of Norway
*The Norwegian language, including th ...
in 2000.
*Honorary D. Sc. from
Rockefeller University in 2001.
*
Longuet-Higgins Prize
The Conference on Computer Vision and Pattern Recognition (CVPR) is an annual conference on computer vision and pattern recognition, which is regarded as one of the most important conferences in its field. According to Google Scholar Metrics (2022 ...
in 2005 and 2009.
*Foreign Member of
The Royal Society in 2008.
*Foreign Member of the
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway.
History
The Royal Frederick Unive ...
.
*Honorary Doctorate from
Brown University
Brown University is a private research university in Providence, Rhode Island. Brown is the seventh-oldest institution of higher education in the United States, founded in 1764 as the College in the English Colony of Rhode Island and Providenc ...
in 2011.
*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
*
Castelnuovo–Mumford regularity
*
Mumford's compactness theorem
*
Haboush's theorem
In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group ''G'' over a field ''K'', and for any linear representation ρ of ''G'' on a ''K''- vector space ''V'', given ''v'' ...
*
Hilbert–Mumford criterion
*
Stable mapping class group
*
Mumford-Tate group
*
Mumford measure
*
Mumford vanishing theorem
*
Theta representation In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heis ...
*
Manin–Mumford conjecture
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has ...
*
Horrocks–Mumford bundle
*
Deligne–Mumford moduli space of stable curves
*
Algebraic stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's re ...
*
Moduli scheme
*
Prym varieties
*
Stable maps
*
Mumford–Shah energy functional
Notes
Publications
* ''Lectures on Curves on Algebraic Surfaces'' (with George Bergman),
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial ...
, 1964.
* ''Geometric Invariant Theory'', Springer-Verlag, 1965 – 2nd edition, with J. Fogarty, 1982; 3rd enlarged edition, with F. Kirwan and J. Fogarty, 1994.
*
* ''Abelian Varieties'',
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print book ...
, 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.
*
Indra's Pearls: The Vision of Felix Klein
* ''Selected Papers on the Classification of Varieties and Moduli Spaces, Springer-Verlag, 2004.
*
*
*
External links
*
*
Mumford's page at Brown University
{{DEFAULTSORT:Mumford, David
1937 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Members of the United States National Academy of Sciences
Fields Medalists
Algebraic geometers
MacArthur Fellows
Putnam Fellows
Brown University faculty
Harvard University faculty
Harvard University alumni
Phillips Exeter Academy alumni
Wolf Prize in Mathematics laureates
Institute for Advanced Study visiting scholars
Foreign Members of the Royal Society
People from Worth, West Sussex
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics
Members of the Norwegian Academy of Science and Letters
Foreign Members of the Russian Academy of Sciences
Members of the American Philosophical Society
Presidents of the International Mathematical Union