David Mumford

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David Mumford
David Mumford.jpg
David Mumford in 2010
Born (1937-06-11) 11 June 1937 (age 88)
Worth, West Sussex, England
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 American mathematician known for his work in algebraic geometry and then for research into 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.

Contents

Life

He was born in Worth, West Sussex in England, of an English father and American mother. His father William Bryant Mumford (born 1900) was educated at Manchester Grammar School and took the Mathematical Tripos at St John's College, Cambridge; [3] as his mother Edith Emily Read had done at Girton College. [4] [5] He started an experimental school in the colonial Tanganyika Territory and at the time of his death in 1951 worked in the United Nations Department of Public Information. [6] [7] He married in 1933 Grace Schiott of Southport, Connecticut, and the couple had five children. [8]

Mumford first went to Unquowa School. [9] He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay-based computer project. [10] [11] 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. [12] He completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. [13]

After a career at Harvard, Mumford moved to Brown University in 1996, moving into applied mathematics. [14] [15]

Research

Algebraic geometry

Mumford's work in geometry combined traditional geometric insights with contemporary algebraic techniques.

Moduli spaces

Mumford published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory.

At the International 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]

Diagram of a fundamental domain (in grey) for the modular group acting on the upper half-plane, the classical picture of moduli for curves of genus 1 ModularGroup-FundamentalDomain.svg
Diagram of a fundamental domain (in grey) for the modular group acting on the upper half-plane, the classical picture of moduli for curves of genus 1

The moduli space of curves of a given genus g was already considered in complex algebraic geometry during the 19th century. When g = 1 the number of moduli is 1 also, the term coming from the elliptic modulus function of the theory of Jacobi elliptic functions. Bernhard Riemann contributed the formula, when g ≥ 2, for the number of moduli, namely 3g−3, a consequence, in deformation theory, of the Riemann-Roch theorem. [17]

The theory of abstract varieties, as Mumford showed, can be applied to provide an existence theorem for moduli spaces of curves, over any algebraically 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 the modular curves, having the predicted dimension. In a summary provided by Jean Dieudonné, making use of scheme theory, the steps are: [18]

  1. Construction of a certain closed subscheme H of the Hilbert scheme of the projective space of dimension n over the ring of integers.
  2. Construction of an "orbit scheme" M of the projective linear group PGLn acting on H. This step relies on the more tractable theory of moduli of abelian varieties.

In the introduction to his 1965 book Geometric 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 an orbit space of an algebraic scheme acted on by an algebraic group exist?". For this area, see algebraic invariant theory. [19] David Gieseker later distinguished, for step #2, going via Torelli's theorem, which was Mumford's original method, and the alternative approach via the Chow point of a curve C, in its projective canonical embedding, which proves the stability of this point. [20]

The concept of stable 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]

Moduli stacks

Before the mathematical concept of stack had been defined, Mumford gave a detailed treatment of the moduli stack of elliptic curves, in his paper Picard Groups of Moduli Problems. In it, he made a calculation of (the analogue of) the stack's Picard group. [22] Hartshorne wrote that, in the paper, Mumford

[...] makes the point that to investigate the more subtle properties of curves, the coarse moduli space may not carry enough information, and so one should really work with stacks. [17]

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 four classes are: [23] [24] [25]

  1. Kodaira dimension minus infinity. These are the ruled surfaces.
  2. Kodaira dimension 0. These are the K3 surfaces, abelian surfaces, hyperelliptic and quasi-hyperelliptic surfaces, and Enriques surfaces. There are classical and non-classical examples in the last two Kodaira dimension zero cases.
  3. Kodaira dimension 1. These are the elliptic and quasi-elliptic surfaces not contained in the last two groups.
  4. Kodaira dimension 2. These are the surfaces of general type.

Explicit equations and algebraic techniques

Mumford wrote on an algebraic approach to the classical theory of theta functions, the theta representation, showing that its algebraic content was large, with resulting finite analogues of the Heisenberg group. [26] This work on the equations defining abelian varieties appeared in 1966–7. He also is one of the founders of the toroidal embedding theory, the geometric approach to varieties defined by monomials. [27] With Dave Bayer he published a paper "What can be computed in algebraic geometry?" in 1993 on computation and algebraic geometry, in which Gröbner basis techniques are fundamental. [28] The concept of Castelnuovo–Mumford regularity has been employed as a complexity measure for Gröbner bases. [29]

Computer vision

In a paper published in 1989, Mumford and Jayant Shah showed that a numerical method could tackle image segmentation, using what became known as the Mumford–Shah functional. In the Handbook of Mathematical Methods in Imaging (2011) the functional was called "classical", and the method described as an energy minimization approach allowing optimal computation for piecewise 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 of Shimon Ullman who brought in variational methods, and Berthold K.P. Horn, Mumford in 1992 addressed the amodal completion problem with concepts from elastica theory. [34] In the 1990s, Song-Chun Zhu at the University of Science and Technology of China, attracted into the field by the book Vision (1982) by David Marr, studied computer science under Mumford. [35]

Promulgation of the ideas of Alexander Grothendieck

In 1962 Mumford was appointed an assistant professor at Harvard. [36] He decided to give a lecture course on a topic in the area of curves on an algebraic surface, to illustrate the importance of the innovative approach in algebraic geometry of Alexander Grothendieck, based at IHES near Paris. [37] The class notes of this course, given 1963–4, were published in 1966 as Lectures on Curves on an Algebraic Surface, and in the Introduction Mumford wrote that the goal was to give an exposition of a "corollary" in Bourbaki seminar #221 given by Grothendieck in early 1961, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV : Les schémas de Hilbert, in the TDTE series. It related to a vexed question in the legacy of the Italian 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 a necessary and sufficient condition for the validity of the theorem was now available. [38] In parallel, Robin Hartshorne was leading a seminar at Harvard, "Residues and Duality", on coherent duality in Grothendieck's derived category setting, in which Mumford, Tate, Stephen Lichtenbaum and others participated. [39]

The Red Book

Mumford's lecture notes on scheme 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." [40] In 1979 in the Bulletin 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." [41] 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. 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 [...] [42]

Grothendieck's relative point of view

In Lectures on Curves on an Algebraic Surface, Mumford wrote "by far the most important categorical notion for algebraic geometry is that of fibre product." [43] Lectures 3 and 4 (with its Appendix on the Zariski 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 as Grothendieck's relative point of view, with geometric illustrations. The representable functor point of view is central: Mumford illustrates it by citing contemporary research: Brown's representability theorem in homotopy theory, where representable functors are clearly distinguished; Tate's rigid analytic spaces; Jaap Murre's work on functors to the category of groups; and Teruhisa Matsusaka's theory of Q-varieties as an extension of the functors considered. [44]

Grothendieck made a case for the "representable functor" approach to replace the universal object conceptualization then prevalent in category 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 a universal bundle with base space a classifying space that represents bundles does not, however, apply directly to moduli. Hence the distinction between a "coarse moduli space", via its geometric points a universal object, and a fine moduli space which represents a functor of all parametrized families of curves. The latter is in line with the "relative point of view". In Geometric 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 role descent morphisms play. [45]

This background explains why faithfully flat descent is introduced in Lecture 4 of Lectures 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 to Riemann's existence theorem and to Grothendieck's existence theorem for formal schemes. [44] Mumford's remark about fibre products may be unpacked in later terms of Grothendieck fibrations and descent theory. [46] [47] He went much further into the relative theory, in defining "modular families" of curves via Grothendieck topologies, in Picard Groups of Moduli Problems. [48]

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 of Michael Artin and others by 1969. There were two sides to this advance. (1) Further innovation in the foundations came with algebraic 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 around artinian rings (commutative and local). See Artin's criterion, formal moduli, Schlessinger's theorem. [49] [50]

Awards and honors

David Mumford in 1975 Mumford2.jpg
David Mumford in 1975

Mumford was awarded a 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; [51] 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 Gisha, an Israeli human rights organization. [52] [53] In 2010 he was awarded the National Medal of Science. [54] In 2012 he became a fellow of the American Mathematical Society. [55]

Further awards and honors include:

Mumford was elected President of the International Mathematical Union in 1995 and served from 1995 to 1999. [51]

Books

Mumford's books Abelian Varieties (with C. P. Ramanujam) and Curves on an Algebraic Surface combined the old and new theories.

See also

Notes

  1. ↑ Mumford, David; Shah, Jayant (1989), "Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems" (PDF), Comm. Pure Appl. Math., XLII (5): 577–685, doi:10.1002/cpa.3160420503
  2. ↑ "Nvidia chief Jensen Huang, film star Tony Leung to get honorary doctorates in Hong Kong", South China Morning Post, 30 October 2024
  3. ↑ Great Britain Colonial (1924), The Colonial Office List, Harrison, p. 699
  4. ↑ "Edith Emily Read (Drage+AND+Girton)", A Cambridge Alumni Database, University of Cambridge
  5. ↑ Kanner, Barbara (1997), Women in Context: Two Hundred Years of British Women Autobiographers, a Reference Guide and Reader, G.K. Hall, p. 609, ISBN   978-0-8161-7346-4
  6. ↑ Fields Medallists' Lectures, World Scientific Series in 20th Century Mathematics, Vol 5, World Scientific, 1997, p. 225, ISBN   978-9810231170
  7. ↑ "British UNO Director Dies", Liverpool Daily Post, 29 January 1951, p. 1
  8. ↑ Burckel, Christian E. (1951), Who's who in the United Nations, vol. 1, C.E. Burckel and Associates., p. 311
  9. ↑ "Beyond the Three Bridges, around the world – Bhāvanā", Bhāvanā The mathematics magazine, 5 (1), January 2021
  10. ↑ "Autobiography of David Mumford", The Shaw Prize, 2006
  11. ↑ David B. Mumford, "How a Computer Works", Radio-Electronics, February 1955, p. 58, 59, 60
  12. ↑ Putnam Competition Individual and Team Winners, Mathematical Association of America, archived from the original on 12 March 2014, retrieved 10 December 2021
  13. ↑ David Mumford at the Mathematics Genealogy Project
  14. ↑ "David Mumford", National Science and Technology Medals Foundation
  15. ↑ Yau, Shing-Tung; Nadis, Steven J. (1 January 2019), The Shape of a Life, Yale University Press, p. 256, ISBN   978-0-300-23590-6
  16. ↑ Lehto, Olli (6 December 2012), Mathematics Without Borders: A History of the International Mathematical Union, Springer Science & Business Media, p. 191, ISBN   978-1-4612-0613-2
  17. 1 2 Hartshorne, Robin (10 December 2009), Deformation Theory, Springer Science & Business Media, p. 177, ISBN   978-1-4419-1595-5
  18. ↑ Dieudonné, Jean (1977), Panorama des mathématiques pures: le choix bourbachique (in French), Gauthier-Villars, p. 141
  19. ↑ Mumford, David (1965), Geometric Invariant Theory, Berlin Heidelberg New York: Springer-Verlag, p. iii
  20. ↑ Mumford, David (15 July 2004), Selected Papers: On the Classification of Varieties and Moduli Spaces, Springer Science & Business Media, p. 3, ISBN   978-0-387-21092-6
  21. ↑ Francoise, Jean-Pierre; Naber, Gregory L.; Tsun, Tsou Sheung (20 June 2006), Encyclopedia of Mathematical Physics, Elsevier Science, p. 45, ISBN   978-0-12-512660-1
  22. ↑ Schilling, O. F. G., ed. (1965), Arithmetical Algebraic Geometry. Proceedings Conference Purdue University, 1963, Harper and Row
  23. ↑ Mumford, David (1969), "Enriques' classification of surfaces in char p I", Global Analysis (Papers in Honor of K. Kodaira), Tokyo: Univ. Tokyo Press, pp. 325–339, doi:10.1515/9781400871230-019, ISBN   978-1-4008-7123-0, JSTOR   j.ctt13x10qw.21, MR   0254053
  24. ↑ Bombieri, Enrico; Mumford, David (1977), "Enriques' classification of surfaces in char. p. II", Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 23–42, MR   0491719
  25. ↑ Bombieri, Enrico; Mumford, David (1976), "Enriques' classification of surfaces in char. p. III." (PDF), Inventiones Mathematicae , 35: 197–232, Bibcode:1976InMat..35..197B, doi:10.1007/BF01390138, MR   0491720, S2CID   122816845
  26. ↑ Mumford, David (15 July 2004), Selected Papers: On the Classification of Varieties and Moduli Spaces, Springer Science & Business Media, p. 293, ISBN   978-0-387-21092-6
  27. ↑ Kempf, G.; Knudsen, F.; Mumford, D.; Saint-Donat, B. (1973), Toroidal Embeddings I
  28. ↑ Eisenbud, David; Robbiano, Lorenzo, eds. (1993), Computational Algebraic Geometry and Commutative Algebra: Cortona 1991, Symposia mathematica, Cambridge University Press, pp. 1–48
  29. ↑ Seiler, Werner M. (26 October 2009), Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra, Springer Science & Business Media, p. 232, ISBN   978-3-642-01287-7
  30. ↑ Mumford, David; Shah, Jayant (July 1989), "Optimal approximations by piecewise smooth functions and associated variational problems", Communications on Pure and Applied Mathematics, 42 (5): 577–685, doi:10.1002/cpa.3160420503
  31. ↑ Scherzer, Otmar, ed. (23 November 2010), Handbook of Mathematical Methods in Imaging, Springer Science & Business Media, p. 1097, ISBN   978-0-387-92919-4
  32. ↑ Montegranario, Hebert (23 December 2024), An Introduction to Variational Calculus: Applications in Image Processing, Springer Nature, p. 127, ISBN   978-3-031-77270-2
  33. ↑ Montegranario, Hebert (23 December 2024), An Introduction to Variational Calculus: Applications in Image Processing, Springer Nature, p. 127, ISBN   978-3-031-77270-2
  34. ↑ Petitot, Jean (September 2003), "Neurogeometry of V1 and Kanizsa Contours", Axiomathes, 13 (3–4): 347–363, doi:10.1023/B:AXIO.0000007240.49326.7e
  35. ↑ Zhu, Song-Chun; Wu, Ying Nian (15 March 2023), Computer Vision: Statistical Models for Marr's Paradigm, Springer Nature, p. xiii, ISBN   978-3-030-96530-3
  36. ↑ Agarwal, Ravi P. (19 October 2025), IMU, ICM, Medals, Prizes, and Laureates, MDPI, ISBN   978-3-7258-4616-0
  37. ↑ Parikh, Carol (23 December 2008), The Unreal Life of Oscar Zariski, Springer Science & Business Media, p. 117 note 153, ISBN   978-0-387-09430-4
  38. ↑ Mumford, David (2 March 2016), Lectures on Curves on an Algebraic Surface, Princeton University Press, pp. vii–viii, ISBN   978-1-4008-8206-9
  39. ↑ Hartshorne, Robin (14 November 2006), Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963 /64, Springer, p. 5, ISBN   978-3-540-34794-1
  40. ↑ Bulletin of the Belgian Mathematical Society, Simon Stevin, vol. 13, Société mathématique de Belgique, 2006, p. 1033
  41. ↑ Bulletin of the American Mathematical Society, vol. 1 (new series), Society, 1979, p. 513
  42. ↑ Mumford, David (11 November 2013), The Red Book of Varieties and Schemes, Springer, p. iv, ISBN   978-3-662-21581-4
  43. ↑ Lectures on Curves on an Algebraic Surface, p.21.
  44. 1 2 Lectures on Curves on an Algebraic Surface, p.18.
  45. ↑ Geometric Invariant Theory, pp. 96–97.
  46. ↑ Grothendieck fibration at the nLab
  47. ↑ descent at the nLab
  48. ↑ Picard Groups of Moduli Problems, p. 52.
  49. ↑ {nLab|id=Artin+representability theorem|title=Artin representability+theorem}}
  50. ↑ Michael Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Princeton Univ. Press, Princeton, N.J., 1969, pp. 21-71. MR0260746 (41:5369)
  51. 1 2 3 4 5 6 7 8 9 O'Connor, John J.; Robertson, Edmund F., "David Mumford", MacTutor History of Mathematics Archive , University of St Andrews
  52. ↑ U.S. prof. gives Israeli prize money to Palestinian university – Haaretz – Israel News, Haaretz, 26 May 2008, retrieved 26 May 2008
  53. ↑ Mumford, David (September 2008), "The Wolf Prize and Supporting Palestinian Education" (PDF), Notices of the American Mathematical Society, 55 (8), American Mathematical Society: 919, ISSN   0002-9920
  54. ↑ Mathematician David Mumford to receive National Medal of Science, Brown University, 15 October 2010, retrieved 25 October 2010
  55. ↑ List of Fellows of the American Mathematical Society, retrieved 2013-02-10.
  56. ↑ "APS Member History", search.amphilsoc.org, retrieved 8 December 2021
  57. ↑ NTNU's list of honorary doctors
  58. ↑ "Chennai Mathematical Institute", www.cmi.ac.in, archived from the original on 22 January 2010, retrieved 7 July 2025
  59. ↑ Gruppe 1: Matematiske fag (in Norwegian), Norwegian Academy of Science and Letters, archived from the original on 10 November 2013, retrieved 7 October 2010
  60. ↑ Commencement 2011: Honorary degrees, 29 May 2011, archived from the original on 15 March 2012, retrieved 29 May 2011


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