View source for David Mumford

{{short description|American mathematician (born 1937)}}
{{for|the Dean of Brechin|David Mumford (priest)}}
{{BLP sources|date=February 2013}}
{{Use dmy dates|date=April 2022}}
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{{Infobox scientist
| name              = David Mumford
| image             = David Mumford.jpg
| caption           = David Mumford in 2010
| birth_date        = {{Birth date and age|1937|06|11|df=y}}
| birth_place       = [[Worth, West Sussex]], [[England]]
| death_date        = 
| death_place       = 
| nationality       = [[Americans|American]]
| field             = [[Mathematics]]
| work_institutions = [[Brown University]]<br />[[Harvard University]]
| alma_mater        = [[Harvard University]]
| thesis_title      = Existence of the moduli scheme for curves of any genus
| thesis_year       = 1961
| doctoral_advisor  = [[Oscar Zariski]]
| doctoral_students = [[Avner Ash]]<br />[[Henri Gillet]]<br />[[Tadao Oda]]<br />[[Emma Previato]]<br />[[Malka Schaps]]<br />[[Michael Stillman]]<br />[[Jonathan Wahl]]<br />[[Song-Chun Zhu]]
| known_for         = [[Algebraic geometry]]<br />[[Mumford surface]]<br />[[Deligne-Mumford stacks]]<br />[[Mumford–Shah functional]]<ref>{{citation
| last1             = Mumford
| given1            = David
| surname2          = Shah
| given2            = Jayant
| title             = Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems
| journal           = Comm. Pure Appl. Math.
| volume            = XLII
| issue = 5
| year              = 1989
| pages             = 577–685
| doi               = 10.1002/cpa.3160420503
| url= https://dash.harvard.edu/bitstream/handle/1/3637121/Mumford_OptimalApproxPiece.pdf?sequence=1
}}</ref>
| prizes            = [[Putnam Fellow]] (1955, 1956)<br />[[Sloan Fellowship]] (1962)<br />[[Fields Medal]] (1974)<br />[[MacArthur Fellowship]] (1987)<br />[[Shaw Prize]] (2006)<br />[[Steele Prize]] (2007)<br />[[Wolf Prize in Mathematics|Wolf Prize]] (2008)<br />[[Longuet-Higgins Prize]] (2005, 2009)<br />[[National Medal of Science]] (2010)<br />[[BBVA Foundation Frontiers of Knowledge Awards|BBVA Foundation Frontiers of Knowledge Award]] (2012)
| honours           = {{ubl|[[Hong Kong University of Science and Technology]] ([[Doctor of Science|Honoris Causa]])<ref>{{citation |title=Nvidia chief Jensen Huang, film star Tony Leung to get honorary doctorates in Hong Kong |url=https://www.scmp.com/news/hong-kong/society/article/3284376/nvidia-chief-jensen-huang-visit-hong-kong-receive-honorary-doctorate-along-film-star-tony-leung |website=South China Morning Post|date=30 October 2024 }}</ref>}}
}}
'''David Bryant Mumford''' (born 11 June 1937) is an American [[mathematician]] known for his work in [[algebraic geometry]] and then for research into [[computer vision|vision]] and [[pattern theory]]. He won the [[Fields Medal]] 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]].

==Early life and education==
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]].<ref name=fml>{{citation|title=Fields Medallists' Lectures, World Scientific Series in 20th Century Mathematics, Vol 5|date=1997|publisher=World Scientific|isbn=978-9810231170|page=225|url=https://books.google.com/books?id=q6eSjV-0egUC&pg=PA225}}</ref>

He attended [[Phillips Exeter Academy]], where he received a [[Westinghouse Science Talent Search]] prize for his relay-based computer project.<ref>"Autobiography of David Mumford", ''The Shaw Prize'', [https://www.shawprize.org/prizes-and-laureates/mathematical-sciences/2006/autobiography-of-david-mumford 2006]</ref><ref>David B. Mumford, "How a Computer Works", ''Radio-Electronics'', February 1955, p. [https://worldradiohistory.com/hd2/IDX-Consumer/Archive-Radio-Electronics-IDX/IDX/50s/1955/Radio-Electronics-1955-02-OCR-Page-0056.pdf 58], [https://worldradiohistory.com/hd2/IDX-Consumer/Archive-Radio-Electronics-IDX/IDX/50s/1955/Radio-Electronics-1955-02-OCR-Page-0057.pdf 59], [https://worldradiohistory.com/hd2/IDX-Consumer/Archive-Radio-Electronics-IDX/IDX/50s/1955/Radio-Electronics-1955-02-OCR-Page-0058.pdf 60]</ref>  Mumford then went to [[Harvard University]], where he became a student of [[Oscar Zariski]]. At Harvard, he became a [[Putnam Fellow]] in 1955 and 1956.<ref name="MMA">{{citation |title=Putnam Competition Individual and Team Winners |url=http://www.maa.org/programs/maa-awards/putnam-competition-individual-and-team-winners |publisher=[[Mathematical Association of America]] |access-date=December 10, 2021 |archive-date=12 March 2014 |archive-url=https://web.archive.org/web/20140312205244/http://www.maa.org/programs/maa-awards/putnam-competition-individual-and-team-winners |url-status=dead }}</ref> He completed his [[PhD]] in 1961, with a thesis entitled ''Existence of the moduli scheme for curves of any genus''.<ref>{{MathGenealogy |id=18755}}</ref>

==Research==
===Algebraic geometry===
Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on [[moduli space]]s, with a theory summed up in his book ''[[Geometric Invariant Theory]]'', on the equations defining an [[abelian variety]], and on [[algebraic surface]]s.  His books ''Abelian Varieties'' (with [[C. P. Ramanujam]]) 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 the time, they were, beside the treatise [[Éléments de géométrie algébrique]], the only accessible introduction. Starting in 1967, the notes were mimeographed, bound in red cardboard, and distributed by Harvard's mathematics department under the title ''Introduction to Algebraic Geometry (Preliminary version of first 3 Chapters)''. Later (1988; 1999, 2nd ed., {{isbn|3-540-63293-X}}), they were published by Springer under the [[Lecture Notes in Mathematics]] series as ''The Red Book of Varieties and Schemes'' (though neither of the two published editions features a red cover).

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

Mumford's research did much to revive the classical theory of [[theta function]]s, 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 is one of the founders of the [[Toric geometry|toroidal embedding]] theory; and sought to apply the theory to [[Gröbner basis]] techniques, through students who worked in algebraic computation.

===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]], 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''====
Mumford's philosophy in characteristic ''p'' was as follows:

<blockquote>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.</blockquote>

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 surface]]s 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]]).  This pathology can now be explained in terms of the [[Picard scheme]] 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]] (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]] is non-zero, but the second Betti number is equal to the rank of the [[Néron–Severi group]]. 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 variety|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.

====Moduli spaces====
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]] in his paper "Murphy's law in algebraic geometry" showed 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 surface]]s 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]] degenerates to a non-reduced group scheme.  The second is the possibility of obtaining [[quasi-elliptic surface]]s 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 surface]]s.
b) Kodaira dimension 0.  These are the [[K3 surface]]s, [[abelian surface]]s, hyperelliptic and [[quasi-hyperelliptic surface]]s, and [[Enriques surface]]s.  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 surface]]s not contained in the last two groups.
d) Kodaira dimension 2. These are the [[surfaces of general type]].

==Awards and honors==
[[File:Mumford2.jpg|thumb|220px|David Mumford in 1975]]
Mumford was awarded a [[Fields medal|Fields Medal]] 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]] for Mathematical Exposition by the [[American Mathematical Society]]. In 2008 he was awarded the [[Wolf Prize]];<ref name=mt>{{MacTutor Biography|id=Mumford}}</ref> on receiving the prize in Jerusalem from [[Shimon Peres]], Mumford announced that he was donating half of the prize money to [[Birzeit University]] in the [[Palestinian territories]] and half to [https://web.archive.org/web/20110726104951/http://www.gisha.org/index.php?intLanguage=2&intSiteSN=137&intItemId=107 Gisha], an Israeli organization that promotes the right to freedom of movement of Palestinians in the Gaza Strip.<ref>{{citation|url=https://www.haaretz.com/1.4984908|title=U.S. prof. gives Israeli prize money to Palestinian university – Haaretz – Israel News|publisher=[[Haaretz]]|date=2008-05-26 |access-date=2008-05-26}}</ref><ref name=notices-ams>{{citation|last=Mumford|first=David|title=The Wolf Prize and Supporting Palestinian Education|journal=Notices of the American Mathematical Society|date=September 2008|volume=55|issue=8|pages=919|url=https://www.ams.org/notices/200808/200808-full-issue.pdf|publisher=American Mathematical Society|issn=0002-9920}}</ref> In 2010 he was awarded the [[National Medal of Science]].<ref>{{citation|url=https://news.brown.edu/pressreleases/2010/10/mumford|title=Mathematician David Mumford to receive National Medal of Science|publisher=[[Brown University]]|date=2010-10-15|access-date=2010-10-25}}</ref> In 2012 he became a fellow of the [[American Mathematical Society]].<ref>[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-02-10.</ref>

There is a long list of awards and honors besides the above, including
*[[Intel Science Talent Search|Westinghouse Science Talent Search]] finalist, 1953.
*[[Harvard Society of Fellows|Junior Fellow]] at Harvard from 1958 to 1961.
*Elected to the [[United States National Academy of Sciences|National Academy of Sciences]] in 1975.<ref name=mt/>
*Honorary Fellow from [[Tata Institute of Fundamental Research]] in 1978.<ref name=mt/>
*Honorary D.Sc. from the [[University of Warwick]] in 1983.<ref name=mt/>
*Foreign Member of [[Accademia dei Lincei|Accademia Nazionale dei Lincei]], [[Rome]], in 1991.<ref name=mt/>
*Honorary Member of [[London Mathematical Society]] in 1995.<ref name=mt/>
*Elected to the [[American Philosophical Society]] in 1997.<ref>{{citation|title=APS Member History|url=https://search.amphilsoc.org/memhist/search?creator=David+Mumford&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced|access-date=2021-12-08|website=search.amphilsoc.org}}</ref>
*Honorary D.Sc. from [[Norwegian University of Science and Technology]] in 2000.<ref>[https://www.ntnu.edu/phd/honorary-doctors NTNU's list of honorary doctors]</ref>
*Honorary D.Sc. from [[Rockefeller University]] in 2001.<ref name=mt/>
*[[Longuet-Higgins Prize]] in 2005 and 2009.
*Honorary Doctorate from [[Chennai Mathematical Institute]] in 2008.<ref>{{Cite web |title=Chennai Mathematical Institute |url=https://www.cmi.ac.in//events/honoriscausa.php |archive-url=https://web.archive.org/web/20100122171806/https://www.cmi.ac.in//events/honoriscausa.php |archive-date=2010-01-22 |access-date=2025-07-07 |website=www.cmi.ac.in}}</ref>
*Foreign Member of [[The Royal Society]] in 2008.<ref name=mt/>
*Foreign Member of the [[Norwegian Academy of Science and Letters]].<ref>{{citation|url=http://www.dnva.no/c26849/artikkel/vis.html?tid=40116 |title=Gruppe 1: Matematiske fag |publisher=[[Norwegian Academy of Science and Letters]] |language=no |access-date=7 October 2010 |url-status=dead |archive-url=https://web.archive.org/web/20131110152102/http://www.dnva.no/c26849/artikkel/vis.html?tid=40116 |archive-date=10 November 2013 }}</ref>
*Honorary Doctorate from [[Brown University]] in 2011.<ref>{{citation|url = http://www.browndailyherald.com/commencement-2011-honorary-degrees-1.2579444|title = Commencement 2011: Honorary degrees|date = 2011-05-29|access-date = 29 May 2011|archive-date = 15 March 2012|archive-url = https://web.archive.org/web/20120315122226/http://www.browndailyherald.com/commencement-2011-honorary-degrees-1.2579444|url-status = dead}}</ref>
*2012 [[BBVA Foundation Frontiers of Knowledge Awards|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.<ref name=mt/>

==See also==
{{div col|colwidth=25em}}
*[[Castelnuovo–Mumford regularity]]
*[[Mumford's compactness theorem]]
*[[Haboush's theorem]]
*[[Hilbert–Mumford criterion]]
*[[Mapping class group#Stable mapping class group|Stable mapping class group]]
*[[Mumford-Tate group]]
*[[Mumford measure]]
*[[Mumford vanishing theorem]]
*[[Theta representation]]
*[[Arithmetic of abelian varieties#Manin–Mumford conjecture|Manin–Mumford conjecture]]
*[[Horrocks–Mumford bundle]]
*[[Deligne–Mumford moduli space of stable curves]]
*[[Algebraic stack]]
*[[Moduli scheme]]
*[[Prym variety|Prym varieties]]
*[[Stable map#The stable map compactification|Stable maps]]
*[[Mumford–Shah functional|Mumford–Shah energy functional]]
{{div col end}}

==Notes==
{{reflist}}

==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.
*{{Citation |last1=Mumford |first1=David |title=The Red Book of Varieties and Schemes |orig-year=1967 (Harvard lecture notes), 1988 (1st ed.) |publisher=[[Springer-Verlag]] |location=Berlin, New York |edition=Expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians, 2nd |series=Lecture Notes in Mathematics |isbn=978-3-540-63293-1 |doi=10.1007/b62130 |mr=1748380 |year=1999 |volume=1358}}
* ''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, [[Emma Previato|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.
*{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | last2=Series | first2=Caroline | last3=Wright | first3=David | title=Indra's Pearls: The Vision of Felix Klein| url=http://klein.math.okstate.edu/IndrasPearls/ | publisher=[[Cambridge University Press]] | isbn=978-0-521-35253-6 | mr=1913879 | year=2002 | doi=10.1017/CBO9781107050051.024}}  [[Indra's Pearls (book)|Indra's Pearls: The Vision of Felix Klein]]
* ''Selected Papers on the Classification of Varieties and Moduli Spaces'', Springer-Verlag, 2004.
*{{citation|mr=2741810 
|last=Mumford|first= David
|title=Selected papers, Volume II. On algebraic geometry, including correspondence with Grothendieck|publisher=Springer|place= New York|year= 2010|isbn= 978-0-387-72491-1}}
*{{citation|last1=Mumford|first1=David|title=Pattern Theory: The Stochastic Analysis of Real-World Signals|first2= Agnès |last2=Desolneux|publisher= A K Peters/CRC Press |year=2010|isbn=978-1568815794|mr=2723182}}
*{{citation|mr=3443857
|last1=Mumford|first1= David|last2=Oda|first2= Tadao
|title=Algebraic geometry. II. 
|series=Texts and Readings in Mathematics|volume= 73|publisher= Hindustan Book Agency|place= New Delhi|year= 2015|isbn= 978-93-80250-80-9|url=http://bookstore.ams.org/hin-70}}
<!--* ''Pattern Theory through Examples ''(with A. Desolneux, in preparation)-->

==External links==
{{commons category}}
{{wikiquote}}

* [https://www.dam.brown.edu/people/mumford/ Mumford's page at Brown University]

{{Fields medalists}}
{{FRS 2008}}
{{Shaw Prize}}
{{Wolf Prize in Mathematics}}

{{Authority control}}

{{DEFAULTSORT:Mumford, David}}
[[Category:1937 births]]
[[Category:Living people]]
[[Category:20th-century American mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:Fields Medalists]]
[[Category:Algebraic geometers]]
[[Category:MacArthur Fellows]]
[[Category:Putnam Fellows]]
[[Category:Brown University faculty]]
[[Category:Harvard University Department of Mathematics faculty]]
[[Category:Harvard University alumni]]
[[Category:Phillips Exeter Academy alumni]]
[[Category:Wolf Prize in Mathematics laureates]]
[[Category:Institute for Advanced Study visiting scholars]]
[[Category:Foreign members of the Royal Society]]
[[Category:People from Worth, West Sussex]]
[[Category:Fellows of the American Mathematical Society]]
[[Category:Fellows of the Society for Industrial and Applied Mathematics]]
[[Category:Members of the Norwegian Academy of Science and Letters]]
[[Category:Foreign members of the Russian Academy of Sciences]]
[[Category:Members of the American Philosophical Society]]
[[Category:Presidents of the International Mathematical Union]]