David Mumford

How to use this course

David Mumford has had two careers. In the first, from 1961 to the early 1980s, he made algebraic geometry able to answer a basic question: how do the shapes of a given kind vary, and what does the space of all of them look like? His geometric invariant theory, his work with Pierre Deligne on stable curves, and his books on abelian varieties made these moduli spaces into concrete objects. In the second, he applied mathematics to vision, perception and the brain, with variational methods and probability as his tools. This course covers both.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Mumford is living, so the biographical lessons cover his career, mathematics, prizes and public roles only.

Timeline

Sussex, Exeter, Harvard, 1937–1957

David Bryant Mumford was born on 11 June 1937 in Worth, Sussex, in the south of England. In 1940 his family moved to the United States. He went to school in Connecticut and then to Phillips Exeter Academy in New Hampshire, where he won a prize in the Westinghouse Science Talent Search for a computer he built from relays.

He entered Harvard in 1953 and was a Putnam Fellow, one of the top scorers in the national undergraduate competition, in 1955 and 1956. At Harvard the lectures of Oscar Zariski drew him to algebraic geometry. Zariski had spent the 1930s and 1940s rebuilding the Italian theory of algebraic surfaces on rigorous commutative algebra. Mumford found in this subject a combination of algebraic precision and geometric intuition that held him for twenty years. He graduated in 1957.

Zariski, Grothendieck and the moduli problem, 1957–1965

Zariski’s seminar at Harvard in the late 1950s was one of the places where the new methods of Jean-Pierre Serre and Alexander Grothendieck entered algebraic geometry. Zariski’s doctoral students of those years included Michael Artin and Heisuke Hironaka as well as Mumford. Mumford absorbed Grothendieck’s theory of schemes, Hilbert schemes and Picard schemes and became one of its most effective users and expositors.

His thesis of 1961 addressed a question going back to Bernhard Riemann, who counted \(3g-3\) complex parameters for curves of genus \(g\). Is there an algebraic variety whose points correspond to isomorphism classes of such curves? Mumford answered by constructing the moduli space as a quotient of a Hilbert scheme by a group action. Making such quotients exist is the subject of geometric invariant theory (Lesson 04).

Also in 1961 he published a short paper in the IHÉS journal on the topology of surface singularities (Lesson 07). He joined the Harvard faculty in 1962 and wrote up Harvard lectures as Lectures on Curves on an Algebraic Surface (1966), with George Bergman, which explained Grothendieck’s Picard scheme through a classical problem.

The road to the Fields Medal, 1965–1974

Geometric Invariant Theory appeared in 1965 in Springer’s Ergebnisse series. Later editions were revised with John Fogarty (1982) and with Fogarty and Frances Kirwan (1994). Mumford became a full professor at Harvard in 1967.

The following years were extraordinarily productive. He developed an algebraic theory of theta functions in a series of papers On the equations defining abelian varieties (1966–67). His lectures at the Tata Institute of Fundamental Research in Bombay became the book Abelian Varieties (1970), still the standard reference. In 1969 he and Pierre Deligne proved that the moduli space of curves of genus \(g\) is irreducible in every characteristic, introducing stable curves and what are now called Deligne–Mumford stacks. He began extending the Enriques classification of surfaces to positive characteristic (1969), a project completed later in the decade with Enrico Bombieri, and with George Kempf, Finn Knudsen and Bernard Saint-Donat developed toroidal embeddings (1973), a tool for resolving singularities and compactifying moduli spaces.

He received the Fields Medal at the International Congress in Vancouver in 1974. John Tate, presenting his work, singled out the theory of moduli and the classification of surfaces in characteristic \(p\). He was elected to the National Academy of Sciences in 1975.

Geometric invariant theory

For the general reader. To build a space of all shapes of some kind, one usually starts with a big space of shapes written in coordinates, then identifies shapes that differ only by a change of coordinates. Dividing by a group in this way can produce something that is not a reasonable space at all. Mumford found which points to throw away so that the quotient is a good geometric space, and he found a simple test for which points to keep.

Affine quotients

Let a reductive group \(G\) (for example \(\mathrm{SL}_n\), \(\mathrm{GL}_n\) or a torus) act on an affine variety \(X=\operatorname{Spec}A\) over \(\mathbb C\). By Hilbert’s theorem \(A^G\) is finitely generated; set \(X/\!\!/G=\operatorname{Spec}A^G\). The map \(X\to X/\!\!/G\) is surjective, and two points have the same image exactly when their orbit closures meet. So \(X/\!\!/G\) parametrizes closed orbits, not all orbits.

Projective quotients

Let \(G\) act on a projective variety \(X\subset\mathbb P^n\) through a linear action on \(\mathbb C^{n+1}\). A point \(x\in X\) with lift \(\hat x\ne0\) is

• semistable if \(0\notin\overline{G\cdot\hat x}\), equivalently some invariant homogeneous polynomial of positive degree is non-zero at \(x\);
• stable if moreover \(G\cdot\hat x\) is closed and the stabilizer of \(\hat x\) is finite;
• unstable otherwise.

So \(x\) is unstable exactly when some one-parameter subgroup drives \(\hat x\) to 0 as \(t\to0\). David Hilbert had found this for \(\mathrm{SL}_n\) acting on forms in 1893; Mumford proved it for all reductive groups and made it the working tool for moduli. Stability of a curve means stability of the point it defines in a Hilbert scheme, and Mumford (1977) and David Gieseker (1982) used this to construct the projective moduli space of stable curves.

Stable curves and their moduli

For the general reader. The space of all smooth curves of a given genus is not compact: a family of smooth curves can degenerate. Deligne and Mumford found the right degenerations to allow, curves with only the simplest crossings and finitely many symmetries. Adding them makes the space compact without spoiling it, and this compact space became one of the central objects of geometry and, later, of string theory.

Dimension

First-order deformations of a smooth curve \(C\) are classified by \(H^1(C,T_C)\), which by Serre duality is dual to \(H^0(C,\omega_C^{\otimes2})\), the space of quadratic differentials. For \(g\ge2\), \(\deg\omega_C^{\otimes 2}=4g-4\gt2g-2\), so Riemann–Roch gives \(h^0=4g-4+1-g=3g-3\), confirming Riemann’s count.

Stable curves

A stable curve of genus \(g\ge2\) is a connected projective curve of arithmetic genus \(g\) whose only singularities are nodes (locally \(xy=0\)), such that every smooth rational component meets the rest of the curve in at least three points. Equivalently, the curve has only nodes and its automorphism group is finite.

Proof idea. Over \(\mathbb C\), irreducibility was known from Teichmüller theory or from Hurwitz spaces. Deligne and Mumford proved properness through the stable reduction theorem: a family of smooth curves over a punctured disc extends, after a finite base change, to a family of stable curves over the whole disc. Since \(\overline{\mathcal M}_g\) is smooth and proper over \(\mathbb Z\), and its fibre over \(\mathbb Q\) is connected, Zariski’s connectedness principle makes every fibre connected; smoothness then makes each fibre irreducible. The paper also defined the algebraic stacks now named after them.

The boundary \(\overline M_g\setminus M_g\) is a union of divisors: \(\Delta_0\), whose general point is an irreducible curve with one node, and \(\Delta_i\) for \(1\le i\le\lfloor g/2\rfloor\), whose general point is two smooth curves of genera \(i\) and \(g-i\) joined at a node.

Abelian varieties and theta functions

For the general reader. An abelian variety is a higher-dimensional version of a torus that can be written by polynomial equations. Its coordinates are given by theta functions, classical series studied by Jacobi and Riemann. Mumford rebuilt the theory of theta functions purely algebraically, so that it works over any field, and used it to write down the equations of abelian varieties and to construct their moduli spaces.

Theta functions

For \(\tau\) in the upper half-plane, Jacobi’s theta function is

It satisfies \(\vartheta(z+1,\tau)=\vartheta(z,\tau)\) and \(\vartheta(z+\tau,\tau)=e^{-\pi i\tau-2\pi i z}\,\vartheta(z,\tau)\), so it is a section of a line bundle on the elliptic curve \(E_\tau=\mathbb C/(\mathbb Z+\tau\mathbb Z)\). In \(g\) variables, with \(\Omega\) in the Siegel upper half-space, the Riemann theta function \(\vartheta(z,\Omega)=\sum_{n\in\mathbb Z^g}\exp(\pi i\,{}^tn\Omega n+2\pi i\,{}^tnz)\) plays the same role for the torus \(\mathbb C^g/(\mathbb Z^g+\Omega\mathbb Z^g)\).

Mumford’s key object is the theta group \(\mathcal G(L^{\otimes n})\) of pairs \((x,\phi)\) with \(x\) an \(n\)-torsion point and \(\phi:L^{\otimes n}\xrightarrow{\sim}t_x^*L^{\otimes n}\). It is a central extension of \(A[n]\cong(\mathbb Z/n)^{2g}\) by \(\mathbb G_m\), a Heisenberg group, and \(H^0(L^{\otimes n})\) is its unique irreducible representation of weight 1. Choosing a basis adapted to this action gives algebraic theta constants, and the Riemann theta relations become identities in the representation. This approach led to his algebraic construction of moduli of abelian varieties with level structure and to the three volumes of Tata Lectures on Theta (1983–1991).

Surfaces: topology, cycles and a fake projective plane

For the general reader. Mumford proved several results about algebraic surfaces that changed what people expected. A singular point that looks smooth to topology must be smooth. The points of a surface can carry far more algebraic information than its topology suggests. And there is a surface with exactly the numbers of the projective plane that is nevertheless completely different from it.

The hypothesis is on the fundamental group. Homology is not enough: the singularity \(x^2+y^3+z^5=0\) has as link the Poincaré homology sphere, whose homology is that of \(S^3\) but whose fundamental group is the binary icosahedral group of order 120. In higher dimensions the analogue fails: John Milnor and Egbert Brieskorn found singular points whose links are topological spheres.

Spencer Bloch conjectured the converse: if \(p_g=0\), then the 0-cycles of degree 0 are controlled by the Albanese variety. Bloch’s conjecture is proved for surfaces of Kodaira dimension below 2 and remains open in general. Mumford’s theorem founded the study of “infinite-dimensional” Chow groups, continued by Roitman, Bloch and Voisin.

A fake projective plane (1979)

Mumford constructed, using 2-adic uniformization and a discrete group acting on the Bruhat–Tits building of \(\mathrm{PGL}_3(\mathbb Q_2)\), a smooth surface of general type with \(K^2=9\) and \(p_g=q=0\). It has the same Betti numbers as \(\mathbb P^2\) but is a ball quotient, not \(\mathbb P^2\). Gopal Prasad and Sai-Kee Yeung later classified the possible arithmetic groups, and Donald Cartwright and Tim Steger completed the list: there are exactly 50 pairs of complex-conjugate fake projective planes.

The enumerative geometry of moduli spaces

For the general reader. Once the space of all curves exists, one can ask about its own shape: its cohomology, and whether it is simple, like projective space, or complicated. In 1982–83 Mumford introduced natural cohomology classes on this space and, with Joe Harris, showed that for large genus it is as far from simple as possible. His conjecture about its cohomology was proved in 2007.

Tautological classes

Let \(\pi:\overline{\mathcal C}_g\to\overline{\mathcal M}_g\) be the universal curve with relative dualizing sheaf \(\omega\). Define the Hodge bundle \(\mathbb E=\pi_*\omega\), of rank \(g\), with Chern classes \(\lambda_i=c_i(\mathbb E)\); the classes \(\kappa_i=\pi_*\big(c_1(\omega)^{i+1}\big)\); and the boundary classes \(\delta_i=[\Delta_i]\), \(\delta=\sum_i\delta_i\). In Towards an enumerative geometry of the moduli space of curves (1983), Mumford applied Grothendieck–Riemann–Roch to \(\pi\) and found, among other relations,

By contrast, Francesco Severi showed that \(M_g\) is unirational for \(g\le10\), and later work extended this to \(g\le14\), so curves of small genus can be written down with free parameters. The proof of general type compares the canonical class \(K_{\overline{\mathcal M}_g}=13\lambda-2\delta_0-3\delta_1-2\delta_2-\cdots-2\delta_{\lfloor g/2\rfloor}\) with the class of an explicit effective divisor, for odd \(g\) the locus of curves with a linear series of unexpectedly small degree, and shows the difference is big. Harris and Mumford also showed that pluricanonical forms extend over the singularities of \(\overline M_g\).

Ib Madsen and Michael Weiss proved this by identifying the stable homology of mapping class groups with that of an infinite loop space, a striking link between algebraic geometry and homotopy theory.

The Mumford–Shah functional

For the general reader. When we look at a scene we see regions separated by edges. A computer receives only a grid of brightness values. Mumford and Jayant Shah proposed to find the regions and edges at once, as the best compromise between three demands: stay close to the image, be smooth inside regions, and use as little edge as possible. Their formula became one of the standard models of image analysis and a hard problem in the calculus of variations.

Let \(\Omega\subset\mathbb R^2\) be the image domain and \(g:\Omega\to\mathbb R\) the observed image. For a closed set \(K\subset\Omega\) (the edges) and a function \(u\) smooth on \(\Omega\setminus K\), the energy is

where \(\mathcal H^1\) is length and \(\mu,\nu\gt0\) are parameters. In the limit where smoothness is strictly enforced, \(u\) is constant on each region and the problem becomes the piecewise-constant model of minimal partitions.

Mumford and Shah showed that the edges of a minimizer, where smooth enough, can only end at crack-tips or meet in triple junctions at 120°, and conjectured that \(K\) is always a finite union of \(C^1\) arcs. The conjecture is open; Alexis Bonnet, Guy David and others have proved it in important special cases.

Pattern theory

For the general reader. Ulf Grenander proposed that perception is inference: the brain combines what it senses with a learned model of what the world is likely to contain. Mumford took up this programme, arguing that the right mathematics for thought is statistics on spaces of structured objects. With students and colleagues he built probability models for textures, contours, natural images and shapes.

Bayes and maximum entropy

Write \(W\) for the state of the world and \(I\) for the image. Pattern theory models \(P(W)\) (a prior over structured objects) and \(P(I\mid W)\) (how images are generated), and infers \(P(W\mid I)\propto P(I\mid W)P(W)\). The Mumford–Shah energy of Lesson 09 is \(-\log\) of such a posterior, up to constants.

Song-Chun Zhu, Ying Nian Wu and Mumford (FRAME, 1997–98) took the \(\phi_k\) to be histograms of responses of an image to a bank of linear filters, fitted the \(\lambda_k\) to match those histograms on training textures, and sampled new textures that look like the originals. With Basilis Gidas, Mumford also studied stochastic models of generic images, built to reproduce the observed scale invariance of natural image statistics. With Peter Michor he studied Riemannian metrics on spaces of curves, showing that the simplest one, the \(L^2\) metric, gives zero geodesic distance, so shape spaces need stronger metrics.

From geometry to vision, 1980–2007

Mumford chaired the Harvard mathematics department from 1981 to 1984 and later held the Higgins professorship. In the early 1980s he turned from algebraic geometry toward the question of how a brain or a computer could see. He has explained the change as a long-standing interest in how thought works, together with the conviction that perception is fundamentally statistical. In 1985 he moved to Harvard’s Division of Applied Sciences. His last major works in algebraic geometry, including the enumerative geometry of moduli and the Harris–Mumford theorem, date from these transitional years.

His work in vision combined variational methods (the Mumford–Shah functional with Jayant Shah; Filtering, Segmentation and Depth with Mark Nitzberg and Takahiro Shiota, 1993) with stochastic models. He worked with Ulf Grenander’s pattern theory group at Brown, and in 1996 moved there as University Professor in the Division of Applied Mathematics. With his former student Tai Sing Lee he proposed a model of the visual cortex as hierarchical Bayesian inference, in which feedback between areas carries predictions. He helped launch Brown’s interdisciplinary brain science programme. His doctoral students in vision and pattern theory include Song-Chun Zhu, Tai Sing Lee, Peter Belhumeur and Mark Nitzberg; in algebraic geometry they include Tadao Oda, Jonathan Wahl, Avner Ash, Henry Pinkham, Ulf Persson, Henri Gillet, Ian Morrison, Ching-Li Chai, Amnon Neeman, Emma Previato and Michael Stillman.

He was a MacArthur Fellow from 1987 to 1992 and twice received the Longuet-Higgins Prize for work in computer vision. He became professor emeritus at Brown.

Public roles, honours and influence

Mumford was President of the International Mathematical Union from 1995 to 1998. He has written for wide audiences: Indra’s Pearls: The Vision of Felix Klein (2002), with Caroline Series and David Wright, explains Kleinian groups and their fractal limit sets through computer experiments, like the picture at the top of this page. The Red Book of Varieties and Schemes, Harvard lecture notes of the 1960s published in 1988, remains a favourite introduction to Grothendieck’s language. He has also written on the history of mathematics, including reviews of the history of Indian mathematics, and on mathematical education. A collection of his essays, Numbers and the World, was published by the American Mathematical Society in 2023.

Honours

• Fields Medal (1974); National Academy of Sciences (1975); Honorary Fellow of the Tata Institute of Fundamental Research (1978).
• Shaw Prize in Mathematical Sciences (2006, with Wu Wenjun), for contributions to mathematics and to pattern theory and vision research.
• AMS Leroy P. Steele Prize for Mathematical Exposition (2007).
• Wolf Prize in Mathematics (2008, with Pierre Deligne and Phillip Griffiths), for algebraic surfaces, geometric invariant theory, and the foundations of the algebraic theory of moduli of curves and theta functions.
• Foreign Member of the Royal Society (2008); U.S. National Medal of Science (2010).
• BBVA Foundation Frontiers of Knowledge Award in Basic Sciences (2012, with Ingrid Daubechies).

Influence

• Moduli: GIT and Deligne–Mumford stacks are the foundation of moduli theory, of Gromov–Witten theory and of the mathematics of string theory, where \(\overline M_{g,n}\) is the space over which amplitudes are integrated.
• Abelian varieties: his books and algebraic theta functions underlie arithmetic geometry, from Faltings’s work to modern computations with Jacobians.
• Vision and imaging: the Mumford–Shah model and its relatives, from active contours to Chan–Vese segmentation, and statistical models of images and shapes.

Connections

People linked to David Mumford across this collection. A name without a link has no page here yet. See David Mumford in the network of all 337 lives →

Teachers and advisors

• Oscar Zariski 1961

Students

• Amnon Neeman 1983
• Avner Ash 1975
• Ching-Li Chai 1984
• Emma Previato 1983
• Finn Knudsen 1972
• Henri Gillet 1978
• Henry Pinkham 1974
• Ian Morrison 1978
• John Fogarty 1966
• Jonathan Wahl 1971
• Mark Nitzberg 1991
• Michael Stillman 1983
• Peter Belhumeur 1993
• Song-Chun Zhu 1996
• Tadao Oda 1967
• Tai Sing Lee 1993
• Ulf Persson 1975

Influenced by

• Alexander Grothendieck
• Felix Klein
• Ulf Grenander

Influenced

• Curtis McMullen

Colleagues and collaborators

• Pierre Deligne 1969
• Agnès Desolneux
• Basilis Gidas
• Caroline Series
• David Wright
• Enrico Bombieri
• Frances Kirwan
• George Kempf 1973
• Ingrid Daubechies
• Jayant Shah 1989
• Joe Harris 1982
• Michael Artin
• Peter Michor
• Phillip Griffiths

Also mentioned in this life

• Leonhard Euler
• Joseph Fourier
• Josiah Willard Gibbs
• Mikhail Gromov
• Werner Heisenberg
• David Hilbert
• Adolf Hurwitz
• Carl Gustav Jacob Jacobi
• Joseph-Louis Lagrange
• Solomon Lefschetz
• John Milnor
• Emmy Noether
• Henri Poincaré
• Bernhard Riemann
• Jean-Pierre Serre
• Carl Ludwig Siegel
• Oswald Teichmüller
• Jacques Tits
• Edward Witten

Fields

• algebraic geometry 17 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, David Bryant Mumford, MacTutor History of Mathematics.
2. David Mumford’s home page at Brown University, with papers, essays and blog.
3. Brown University, Mumford awarded the National Medal of Science (2010).
4. Brown University, Mumford wins Wolf Prize (2008).
5. BBVA Foundation, Frontiers of Knowledge Award: David Mumford (2012).
6. Mathematics Genealogy Project, David Bryant Mumford.
7. International Mathematical Union, Executive Committee 1995–1998.
8. D. Mumford, Numbers and the World: Essays on Math and Beyond, AMS, 2023.

Mathematics

1. D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, 3rd ed., Springer, 1994.
2. P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHÉS 36 (1969), on Numdam.
3. D. Mumford, Abelian Varieties, Tata Institute / Oxford University Press, 1970; 2nd ed. 1974.
4. D. Mumford, On the equations defining abelian varieties I–III, Invent. Math. 1 and 3 (1966–67).
5. D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. IHÉS 9 (1961).
6. D. Mumford, Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 (1969).
7. D. Mumford, An algebraic surface with K ample, K²=9, p_g=q=0, Amer. J. Math. 101 (1979).
8. J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982).
9. D. Mumford, Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry II, Birkhäuser, 1983.
10. G. Farkas, D. Jensen, S. Payne, The Kodaira dimensions of \(\overline{\mathcal M}_{22}\) and \(\overline{\mathcal M}_{23}\), arXiv:2005.00622.
11. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989).
12. D. Mumford and A. Desolneux, Pattern Theory: The Stochastic Analysis of Real-World Signals, A K Peters, 2010.
13. D. Mumford, C. Series, D. Wright, Indra’s Pearls: The Vision of Felix Klein, Cambridge, 2002.
14. D. Mumford, The Red Book of Varieties and Schemes, Springer, 1988; 2nd ed. 1999.