David Mumford | |
|---|---|
 David Mumford in 2010 | |
| Born | (1937-06-11)11 June 1937 (age 88) |
| Alma mater | Harvard University |
| Known for | Algebraic geometry Mumford surface Deligne-Mumford stacks Mumford–Shah functional[1] |
| Awards | Putnam Fellow (1955, 1956) Sloan Fellowship (1962) Fields Medal (1974) MacArthur Fellowship (1987) Shaw Prize (2006) Steele Prize (2007) Wolf Prize (2008) Longuet-Higgins Prize (2005, 2009) National Medal of Science (2010) BBVA Foundation Frontiers of Knowledge Award (2012) |
| Honours | |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Brown University Harvard University |
| Thesis | Existence of the moduli scheme for curves of any genus (1961) |
| Doctoral advisor | Oscar Zariski |
| Doctoral students | Avner Ash Henri Gillet Tadao Oda Emma Previato Malka Schaps Michael Stillman Jonathan Wahl Song-Chun Zhu |
David Bryant Mumford (born 11 June 1937) is an Americanmathematician known for his work inalgebraic geometry and then for research intovision andpattern theory. He won theFields Medal and was aMacArthur Fellow. In 2010 he was awarded theNational Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics atBrown University.
He was born inWorth, West Sussex inEngland, of an English father and American mother. His father William Bryant Mumford (born 1900) was educated atManchester Grammar School and took theMathematical Tripos atSt John's College, Cambridge;[3] as his mother Edith Emily Read had done atGirton College.[4][5] He started an experimental school in the colonialTanganyika Territory and at the time of his death in 1951 worked in theUnited Nations Department of Public Information.[6][7] He married in 1933 Grace Schiott ofSouthport, Connecticut, and the couple had five children.[8]
Mumford first went toUnquowa School.[9] He attendedPhillips Exeter Academy, where he received aWestinghouse Science Talent Search prize for his relay-based computer project.[10][11] Mumford then went toHarvard University, where he became a student ofOscar Zariski. At Harvard, he became aPutnam Fellow in 1955 and 1956.[12] He completed hisPhD in 1961, with a thesis entitledExistence of the moduli scheme for curves of any genus.[13]
After a career at Harvard, Mumford moved toBrown University in 1996, concentrating onapplied mathematics.[14][15]
Mumford's work in geometry combined traditional geometric insights with contemporary algebraic techniques.
Mumford published onmoduli spaces, with a theory summed up in his bookGeometric Invariant Theory.
At theInternational Congress of Mathematicians in 1974,John Tate said:
Mumford's major work has been a tremendously successful multi-pronged attack on problems of the existence and structure of varieties of moduli, that is, varieties whose points parametrize isomorphism classes of some type of geometric object.[16]
Themoduli space of curves of a givengenusg was already considered incomplex algebraic geometry during the 19th century. Wheng = 1 the number of moduli is 1 also, the term coming from theelliptic modulus function of the theory ofJacobi elliptic functions.Bernhard Riemann contributed the formula, wheng ≥ 2, for the number of moduli, namely 3g−3, a consequence, indeformation theory, of theRiemann-Roch theorem.[17]
The theory ofabstract varieties, as Mumford showed, can be applied to provide anexistence theorem for moduli spaces of curves, over anyalgebraically closed field. It therefore answers the question of in what sense there is a geometric object, with an algebraic definition, that parametrizes algebraic curves, so generalizing themodular curves, having the predicted dimension. In a summary provided byJean Dieudonné, making use ofscheme theory, the steps are:[18]
In the introduction to his 1965 bookGeometric Invariant Theory, Mumford described the construction of "moduli schemes of various types of objects" as essentially "a special and highly non-trivial case" of the problem "when does anorbit space of analgebraic scheme acted on by analgebraic group exist?". For this area, seealgebraic invariant theory.[19]David Gieseker later distinguished, for step #2, going viaTorelli's theorem, which was Mumford's original method, and the alternative approach via theChow point of a curveC, in its projectivecanonical embedding, which proves thestability of this point.[20]
The concept ofstable vector bundle from moduli theory has been consequential in mathematical physics: "This is a mathematical notion of stability, but it also corresponds to physical stability, at least in a regime in which quantum corrections are small."[21]
Before the mathematical concept ofstack had been defined, Mumford gave a detailed treatment of themoduli stack of elliptic curves, in his paperPicard Groups of Moduli Problems. In it, he made a calculation of (the analogue of) the stack'sPicard group.[22] Hartshorne wrote that, in the paper, Mumford
[...] makes the point that to investigate the more subtle properties of curves, thecoarse moduli space may not carry enough information, and so one should really work with stacks.[17]
In three papers written between 1969 and 1976 (the last two in collaboration withEnrico Bombieri), Mumford extended theEnriques–Kodaira classification of smoothprojective surfaces from the case of the complexground field to the case of analgebraically closed ground field of characteristicp. The four classes are:[23][24][25]
Mumford wrote on an algebraic approach to the classical theory oftheta functions, thetheta representation, showing that its algebraic content was large, with resulting finite analogues of theHeisenberg group.[26] This work on theequations defining abelian varieties appeared in 1966–7. He also is one of the founders of thetoroidal embedding theory, the geometric approach to varieties defined bymonomials.[27] WithDave Bayer he published a paper "What can be computed in algebraic geometry?" in 1993 on computation and algebraic geometry, in whichGröbner basis techniques are fundamental.[28] The concept ofCastelnuovo–Mumford regularity has been employed as a complexity measure for Gröbner bases.[29]
In a paper published in 1989, Mumford and Jayant Shah showed that a numerical method could tackleimage segmentation, using what became known as theMumford–Shah functional. In theHandbook of Mathematical Methods in Imaging (2011) the functional was called "classical", and the method described as anenergy minimization approach allowing optimal computation forpiecewise function solutions.[30][31][32] Another assessment in 2024 was "a foundational approach for partitioning an image into meaningful regions while preserving edges."[33]
Following work ofShimon Ullman who brought in variational methods, andBerthold K.P. Horn, Mumford in 1992 addressed theamodal completion problem with concepts fromelastica theory.[34] In the 1990s,Song-Chun Zhu at theUniversity of Science and Technology of China, attracted into the field by the bookVision (1982) byDavid Marr, studied computer science under Mumford.[35]
In 2002, Mumford and Tai Sing Lee proposed a model for thevisual cortex, influenced by work onpattern theory byUlf Grenander.[36][37] The book chapter "The Evolving Concept of Cortical Hierarchy" from 2023 by Vezoli, Hou and Kennedy states that Mumford's "Bayesian computational approach to cortical hierarchy was revolutionary".[38] The 2023 book by Zhu and Wu,Computer Vision: Statistical Models for Marr's Paradigm, in its Preface lists as "pioneers of vision"Béla Julesz andKing-Sun Fu, with Grenander, Maar and Mumford.[39]
In 1962 Mumford was appointed an assistant professor at Harvard.[40] He decided to give a lecture course on a topic in the area of curves on analgebraic surface, to illustrate the importance of the innovative approach in algebraic geometry ofAlexander Grothendieck, based atIHES near Paris.[41] The class notes of this course, given 1963–4, were published in 1966 asLectures on Curves on an Algebraic Surface, and in the Introduction Mumford wrote that the goal was to give an exposition of a "corollary" inBourbaki seminar #221 given by Grothendieck in early 1961,Techniques de construction et théorèmes d'existence en géométrie algébrique. IV : Les schémas de Hilbert, in theTDTE series. It related to a vexed question in the legacy of theItalian school of algebraic geometry, called the "completeness of the characteristic system" of a suitable algebraic system of curves. The issue was of a possible "pathology", and the burden of the course was to show that anecessary and sufficient condition for the validity of the theorem was now available.[42] In parallel,Robin Hartshorne was leading a seminar at Harvard, "Residues and Duality", oncoherent duality in Grothendieck'sderived category setting, in which Mumford, Tate,Stephen Lichtenbaum and others participated.[43]
Mumford's lecture notes onscheme theory circulated for years in unpublished form. A geometric motivation for the theory brought forward by Mumford was "infinitesimals well adapted to algebraic and geometric structures."[44] In 1979 in theBulletin of the American Mathematical Society a reviewer wrote of this work that it "initiated a generation of students to the subject when nothing else was available."[45] At the time, they were, beside the treatiseÉléments 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 titleIntroduction 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 theLecture Notes in Mathematics series asThe Red Book of Varieties and Schemes. In an introduction to the 1988 edition, Mumford wrote
[...] Grothendieck came along and turned a confused world of researchers upside down, overwhelming them with the new terminology of schemes as well as with a huge production of new and very exciting results. These notes attempted to show something that was still very controversial at that time: that schemes really were the most natural language for algebraic geometry [...][46]
InLectures on Curves on an Algebraic Surface, Mumford wrote "by far the most important categorical notion for algebraic geometry is that offibre product."[47] Lectures 3 and 4 (with its Appendix on theZariski tangent space) were a concise introduction to the circle of ideas, based on scheme theory (the book uses the older terminology of pre-schemes) known asGrothendieck's relative point of view, with geometric illustrations. Therepresentable functor point of view is central: Mumford illustrates it by citing contemporary research:Brown's representability theorem inhomotopy theory, where representable functors are clearly distinguished; Tate'srigid analytic spaces;Jaap Murre's work on functors to thecategory of groups; andTeruhisa Matsusaka's theory of Q-varieties as an extension of the functors considered.[48]
Grothendieck made a case for the "representable functor" approach to replace theuniversal object conceptualization then prevalent incategory theory. For moduli spaces, the classical idea of a "universal family of curves" would therefore be modified to a functor "of curves", with a proof that it was representable. The satisfactory picture from homotopy theory, of auniversal bundle with base space aclassifying space that represents bundles does not, however, apply directly to moduli. Hence the distinction between a "coarse moduli space", via itsgeometric points a universal object, and afine moduli space which represents a functor of all parametrized families of curves. The latter is in line with the "relative point of view". InGeometric Invariant Theory Mumford laid out the issues for moduli theory, stating that currently "one knows very few ways in which to pose a plausible fine moduli problem." He went on to state that in technical terms the fine moduli problem was not "essentially harder", given "descent machinery", a reference to the roledescent morphisms play.[49]
This background explains whyfaithfully flat descent is introduced in Lecture 4 ofLectures on Curves on an Algebraic Surface: as a necessary condition for representability of a functor. Mumford names the representability issue for schemes "Grothendieck's existence problem", a nod toRiemann's existence theorem and to Grothendieck's existence theorem forformal schemes.[48] Mumford's remark about fibre products may be unpacked in later terms ofGrothendieck fibrations anddescent theory.[50][51] He went much further into the relative theory, in defining "modular families" of curves viaGrothendieck topologies, inPicard Groups of Moduli Problems.[52]
The lacuna of the mid-1960s theory in terms of sufficient conditions for representability of functors, as would apply to moduli problems, was filled through work ofMichael Artin and others by 1969. There were two sides to this advance. (1) Further innovation in the foundations came withalgebraic spaces (more general than schemes but more restricted than Matsusaka's Q-varieties or Mumford's stacks). Algebraic spaces allow more quotients by equivalence relations than schemes. (2) This definition was combined with a thorough use of algebraic infinitesimal techniques aroundartinian rings (commutative andlocal). SeeArtin's criterion,formal moduli,Schlessinger's theorem.[53][54]
Mumford was awarded aFields Medal in 1974. He was aMacArthur Fellow from 1987 to 1992. He won theShaw Prize in 2006. In 2007 he was awarded theSteele Prize for Mathematical Exposition by theAmerican Mathematical Society. In 2008 he was awarded theWolf Prize;[55] on receiving the prize in Jerusalem fromShimon Peres, Mumford announced that he was donating half of the prize money toBirzeit University in thePalestinian territories and half toGisha, an Israeli human rights organization.[56][57] In 2010 he was awarded theNational Medal of Science.[58] In 2012 he became a fellow of theAmerican Mathematical Society.[59]
Further awards and honors include:
Mumford was elected President of theInternational Mathematical Union in 1995 and served from 1995 to 1999.[55]
Mumford's booksAbelian Varieties (withC. P. Ramanujam) andCurves on an Algebraic Surface combined the old and new theories.
