Donald C. Spencer (1912–2001)

Joseph J. Kohn, Phillip A. Griffiths, Hubert Goldschmidt, Enrico Bombieri, Bohous Cenkl, Paul Garabedian, Louis Nirenberg

Joseph J. Kohn on sequences of integers; in applied on Donald C. Spencer died on December 23, 2001. He was fluid mechanics; and in the born on April 25, 1912, in Boulder, Colorado. He was theory of one complex vari- an undergraduate at the University of Colorado (B.A. able on univalent functions, in 1934) and at the Massachusetts Institute of Tech- on conformal mappings, nology (B.S. in Aeronautical Engineering in 1936) and and on Riemann surfaces. a graduate student in mathematics at Cambridge At Princeton, Spencer’s University (Ph.D. in 1939, Sc.D. in 1963). His doctoral research turned to several thesis, “On a Hardy-Littlewood problem of dio- complex variables and com- phantine approximation and its generalizations”, plex manifolds. He collabo- was written under the direction of J. E. rated with Kunihiko Littlewood and initiated Spencer’s remarkable math- Kodaira, who is also widely Donald C. Spencer ematical career. Spencer taught at M.I.T. (1939–42), recognized as one of the at Stanford (1942–50 and 1963–68), and at Princeton great mathematical pioneers of the twentieth cen- (1950–63 and 1968–78). He was the Eugene Higgins tury. Together, Spencer and Kodaira developed the Professor at Princeton (1971–72) and the Henry modern theory of deformations of complex mani- Burchard Fine Professor at Princeton (1972–78). He folds into a major tool, the basis for a large body received the degree Sc.D. honoris causa from of subsequent research. This work has had and Purdue University (1971) and was the joint recipi- continues to have tremendous influence in large seg- ent with A. C. Schaeffer of the Bôcher Memorial Prize ments of mathematics, including the theory of sev- (1948). He was a member of the U.S. National Acad- eral complex variables, , al- emy of Sciences (from 1961), a fellow of the Ameri- gebraic geometry, and mathematical . In can Academy of Arts and Sciences (from 1967), and fact, Spencer’s work with Kodaira was one of the a recipient of the National Medal of Science (1989). most remarkable mathematical collaborations of the He was the author of numerous important research twentieth century; its only parallel is the famous papers and four influential books (see [1], [2], [3], Hardy-Littlewood work. To give a feeling for the ex- [4], [5]). citement generated by these efforts, I quote from Spencer’s mathematical work is truly impressive the introduction to Kodaira’s book (Complex and spans many fields in which he made funda- Manifolds and Deformations of Complex Structures, mental contributions. Before coming to Princeton, Springer-Verlag, 1981): he worked in number theory on lattice points and In order to clarify this mystery, Spencer and I developed the theory of defor- Joseph J. Kohn is professor of mathematics at Princeton University. His email address is [email protected]. mations of compact complex manifolds.

JANUARY 2004 NOTICES OF THE AMS 17 Just as Spencer had an unfailing instinct for how to approach mathematical research, so he also had an unfailing instinct for how to inspire both stu- dents and fellow . Recently Victor Guillemin, professor of mathematics at M.I.T., wrote the following tribute: When I first met Don Spencer, I was a twenty-two-year-old graduate student, and Spencer was in his mid-forties and at the height of his fame. Kodaira-Spencer was the most exciting development of that era in differential geometry, the early sixties equivalent of Seiberg-Witten, and I and a lot of my fellow graduate students were caught up in this excitement and After retirement in Durango, CO. working on Spencer sequences, Spencer The process of the development was cohomology, the Spencer theory of de- the most interesting experience in my formations, etc. This was the state of whole mathematical life. It was similar affairs when I first met Don in the fall of to an experimental science developed by 1960; and therefore, not surprisingly, the the interaction between experiments thing I most remember about him was (examination of examples) and theory. his incredible kindness and empathy for In this book I have tried to reproduce young people. I have had several older this interesting experience; however I colleagues who have been able, as one could not fully convey it. Such an ex- says, to “bridge the generation gap”, but perience may be a passing phenomenon there is no one I’ve ever known who had which cannot be reproduced. this quality to the extent that Spencer did. Some of us took advantage of it and Furthermore, in 1987 Kodaira wrote in his let- must at times have bored him silly with ter to the President’s Committee on the National the catalogue of our accomplishments Medal of Science: and aspirations. But he was unfailing non- Spencer’s contributions to mathematics judgmental and sympathetic. More than go far beyond his published papers. He that, one could feel that his interest in exerted tremendous influence on his col- these aspirations was the genuine com- laborators and students. His enthusiasm modity, and not just feigned to make us knew no limit and was contagious. In feel good. One’s ego at 22 is a fragile Princeton he was always surrounded by vessel, and Don instinctively knew this. a group of mathematicians who shared For that (as for much else) I remember his enthusiasm and collaborated in the Don as the kindest, nicest person I’ve research of complex analysis (I was one ever known. of them). In the 1950s the theory of com- In 1978 Spencer retired from Princeton and plex manifolds was developed extensively moved to Durango, Colorado. There he became in Princeton. The driving force behind very active in the conservation and ecology move- this development was in fact Spencer’s ments and also an avid hiker. He soon made many enthusiasm. friends and became well known in the area. The city At the same time, Spencer introduced the use of Durango designated April 25 as “Don Spencer of potential theory in the study of complex mani- Day”. After his retirement, Don Spencer kept in folds with boundaries, and in particular formu- touch with his many friends and colleagues, al- lated the “d-bar-Neumann problem”, which has led ways inspiring, supportive, and full of enthusiasm. to very important developments in both several Don was my thesis adviser, mentor, colleague, complex variables and partial differential equa- and friend. I consider myself fortunate to have tions. In the last decade of his career, Spencer been so close to such a remarkable human being. worked on overdetermined systems of partial dif- Here I will give a brief account of his mathemati- ferential equations and on pseudogroups. cal contributions to the theory of partial differen- In all his work Spencer shows remarkable origi- tial equations (P.D.E.). nality and insight. His influence on his students, his Don’s contributions to P.D.E. can be divided collaborators, and his many friends also has had a into three groups: (1) equations arising from fluid lasting impact on twentieth-century mathematics. mechanics, (2) equations arising from complex

18 NOTICES OF THE AMS VOLUME 51, NUMBER 1 analysis, and (3) the theory of linear overdeter- boundary-value problem mined systems. His work in (1) is discussed below for an elliptic operator by Paul Garabedian. The work in (2) is divided into (the boundary values are two parts: equations arising from one complex imposed by the require- variable and those from several complex variables. ment of belonging∗ to the Here I will concentrate on the work connected with domain of ∂ ). Spencer several complex variables. The contributions in (3) and I (see [10]) studied are discussed by Hubert Goldschmidt. this phenomenon by in- The development of the theory of P.D.E. is closely troducing the operator linked with advances in complex analysis; in fact, ∂τ = ∂ + τ∂, setting up Riemann’s approach to the study of conformal the analogous ∂τ -Neu- mapping via the Dirichlet principle led to the sys- mann problem, and tematic development of the theory of elliptic P.D.E. studying its behavior for and associated variational problems. The applica- small τ. In [10] we man- tion of these methods to the theory of several com- aged to solve the problem plex variables was initiated by Hodge in his theory on balls in Cn by use of of harmonic integrals on compact manifolds. It is spherical harmonics, but this work that led H. Weyl to prove the funda- this case is too special for mental hypoellipticity theorem, known as Weyl’s the important applica- lemma, which in turn led to the development of the tions. general theory of elliptic P.D.E. One of the main appli- The theory of harmonic integrals on compact cations of the ∂-Neumann manifolds is a crucial ingredient in the Kodaira- problem on (0, 1)-forms, Receiving National Medal of Science Spencer theory of deformations of complex envisioned by Spencer, is from President George Bush, 1989. structures. An example of this is the fundamental the construction of holo- existence theorem proved by Kodaira, Nirenberg, morphic functions with specified properties. It is and Spencer (see [6]), which is described below by this approach that C. B. Morrey used in [11] to P. A. Griffiths. show that a compact real-analytic manifold M can Spencer’s most spectacular contribution to the be embedded real analytically into euclidean space. theory of P.D.E., one which has been a major in- Thus Morrey wanted to use the ∂τ -Neumann prob- fluence in mathematical research, is setting down lem with Ω a thin tubular neighborhood of M in the program to generalize harmonic integrals to the complexification of M; if the ∂τ -Neumann prob- noncompact manifolds. Spencer called this pro- lem could be solved on this Ω, then enough inde- gram the ∂ -Neumann problem. Spencer’s contri- pendent holomorphic functions could be con- butions were not simply confined to his papers: structed to give the required embedding. In [11] 1 they were also made through his teaching, lec- Morrey proved the following fundamental “2 esti- tures, and principally his unique remarkable mate”. There exists a constant C>0 such that ability to communicate his ideas and his enthusi- ∗ 2 2 2 2 asm to so many students and colleagues. His first ϕ 1 ≤ C( ∂ϕ + ∂ ϕ + ϕ ) papers in the direction of the ∂ -Neumann problem 2 were [7] with G. F. D. Duff and [8] with P. R. Garabe- for all (0, 1)-forms ϕ whose coefficients∗ are in dian. The ∂ -Neumann problem for (p, q)-forms C∞(Ω) and which are in the domain of ∂ . Here on a manifold Ω can be formulated as follows. Let · 1 1 denotes the Sobolev 2-norm. Morrey’s proof Lp,q(Ω) denote the space of square-integrable (p, q)- 2 2 of this estimate works for any complex manifold p,q → p,q+1 forms on Ω, let ∂ : L2 (Ω) L2 (Ω) denote the with smooth strongly pseudoconvex boundary. ∗ − L closure of ∂ , and let ∂ : Lp,q(Ω) → Lp,q 1(Ω) de- 1 2 2 2 Using the “2 estimate”, I proved (see [12]) that note the L2 adjoint of ∂ . The ∂ -Neumann problem the solution of the ∂ -Neumann problem exists for ∈ p,q ∈ p,q is: given α L2 (Ω), does there exist a ϕ L2 (Ω) manifolds with strongly pseudoconvex smooth that∗ is in the intersection of the domains∗ ∗ of ∂ and boundaries and that, locally near the boundary, of ∂ with ∂ϕ in the domain of ∂ and ∂ ϕ in the the solution ϕ gains one derivative on α (in the domain of ∂ such that interior it automatically gains two derivatives). ∗ ∗ Hörmander, in [13], proved existence using a dif- ∂ ∂ ϕ + ∂ ∂ϕ = α? ferent approach. He used Morrey’s techniques to In [9] Spencer formulated this problem for the obtain L2 estimates of ϕ with weights that are sin- case when Ω ⊂ X, where X is a complex manifold, gular at the boundary. In this way he was able to Ω is compact, and Ω has a smooth boundary, and prove existence in L2 without assuming smoothness he outlined some of the important applications. The of the boundary or strong pseudoconvexity; all main feature of the ∂ -Neumann problem on man- that is needed is weak pseudoconvexity. Because ifolds with boundary is that it is a nonelliptic of interior ellipticity, Hörmander’s solution gains

JANUARY 2004 NOTICES OF THE AMS 19 two derivatives locally in the interior, but there is [8] P. R. GARABEDIAN and D. C. SPENCER, A complex tensor no control near the boundary. calculus for Kähler manifolds, Acta Math. 89 (1953), The ∂ -Neumann problem and the mathematics 279–331. that grew out of it have had an enormous impact [9] D. C. SPENCER, Les opérateurs de Green et Neumann sur on the theory of several complex variables, on the les variétés ouvertes riemanniennes et hermitiennes, lectures given at the Collège de France, 1955 (copy in theory of P.D.E., on analysis, and recently on alge- the library of the Institut Henri Poincaré). braic geometry. In particular, Hörmander’s solution [10] J. J. KOHN and D. C. SPENCER, Complex Neumann prob- has engendered “the L2 methods”, which have been lems, Ann. of Math. 66 (1957), 89–140. a very powerful tool. Hörmander has recently writ- [11] C. B. MORREY, The analytic embedding of abstract ten a paper (see [14]), dedicated to the memory of real analytic manifolds, Ann. of Math. 68 (1958), D. C. Spencer, which exposes this approach and its 159–201. consequences. The approach, based on regularity [12] J. J. KOHN, Harmonic integrals on strongly pseudo- near the boundary, has led to numerous develop- convex manifolds. I, Ann. of Math. 78 (1963), 112–148; II, Ann. of Math. 79 (1964), 450–472. ments. Some of these are the study of subelliptic 2 [13] L. HÖRMANDER, L estimates and existence theorems estimates; the calculus of pseudodifferential for the ∂ operator, Acta Math. 113 (1965), 89–152. operators; the study of the operators ∂b and b [14] ——— , A history of existence theorems for the Cauchy- on CR manifolds and on the Heisenberg group; Riemann complex in L2 spaces, J. Geom. Anal. 13 and analysis on weakly pseudoconvex manifolds: (2003), 307–335. global regularity and irregularity, regularity of bi- holomorphic and proper mappings, multiplier ideals, etc. Multiplier ideals, which arose from the Phillip A. Griffiths study of estimates for the ∂-Neumann problem, are Don Spencer and his friend and collaborator currently used effectively in Kähler geometry, Kunihiko Kodaira were the principal founders of algebraic geometry, and the study of degenerate deformation theory—a central part of modern al- elliptic P.D.E. gebraic geometry and indeed of modern mathe- Don played an essential role in all this during matics. The development of Kodaira-Spencer the- the 1950s and 1960s. Apart from his own contri- ory, as it is now called, took place during the middle butions, he communicated his ideas and enthusi- and late 1950s. With the benefit of hindsight we asm to most mathematicians who worked on these may see it as arising from a natural confluence of questions, students and colleagues alike. These in- Spencer’s previous interests and some of the major cluded: A. Andreotti, M. Ash, P. E. Conner, mathematical developments at the time. H. Grauert, L. Hörmander, S. H. C. Hsiao, N. Kerz- On the one hand, through his work with Schaeffer man, J. J. Kohn, M. Kuranishi, C. B. Morrey, L. Niren- [17] and Schiffer [18], [19], [20], Spencer was inti- berg, H. Rossi, and W. J. Sweeney. All of us found mately familiar with moduli of Riemann surfaces: Don ready at any time to listen to us, encourage firstly the by-then-classical theory of Teichmüller us, challenge us, and make us appreciate the im- for compact Riemann surfaces and secondly the portance of the enterprise. In the 1970s, 1980s, and more general case where the Riemann surface may 1990s the number of people working on these have a boundary (to which Spencer and his collab- questions increased dramatically, and I am sure that orators made significant contributions). Central to Don had a direct influence on many of these also. the former theory are the quadratic differentials on the Riemann surface. As will be seen below, the References search for their higher-dimensional analogues was [1] D. C. SPENCER, Selecta, three volumes, World Scientific, a significant issue. 1985. On the other hand, three aspects of the general [2] A. C. SCHAEFFER and D. C. SPENCER, Coefficient Regions mathematical environment of the time were partic- for Schlicht Functions, Amer. Math. Soc. Colloq. Publ., ularly relevant. Namely, this was a period of intense vol. 35, 1950. activity in the areas of [3] M. SCHIFFER and D. C. SPENCER, Functionals of Finite (i) harmonic integrals, Riemann Surfaces, Princeton Univ. Press, 1954. (ii) sheaf theory, and [4] H. K. NICKERSON, N. E. STEENROD, and D. C. SPENCER, (iii) several complex variables. Advanced Calculus, Van Nostrand, 1959. The first was in the tradition of Riemann, Hodge, [5] A. KUMPERA and D. C. SPENCER, Lie Equations. Volume I: General Theory, Ann. of Math. Stud., No. 73, Prince- and Weyl, among others, and was a subject to ton Univ. Press, 1972. which each of Kodaira and Spencer individually [6] K. KODAIRA, L. NIRENBERG, and D. C. SPENCER, On the made significant contributions (cf. [1], [2], [3]). existence of deformations of complex analytic Much of this work was concerned with what is now structures, Ann. of Math. 68 (1958), 450–459. the “standard” linear elliptic theory on compact [7] G. F. D. DUFF and D. C. SPENCER, Harmonic tensors on manifolds with boundary, Proc. Nat. Acad. Sci. U.S.A. Phillip A. Griffiths is director of the Institute for Advanced 37 (1951), 614–619. Studies, Princeton. His email address is [email protected].

20 NOTICES OF THE AMS VOLUME 51, NUMBER 1 manifolds, including those with boundary, and also be reflected in the two equivalent approaches especially the Kähler case. to Kodaira-Spencer’s deformation theory. The second was a very general theory, which for Before turning to that theory, I would call at- present purposes may be thought of as providing tention to the important series of papers [7], [8], a systematic framework for analyzing the ob- [6], [9] by Kodaira and Spencer and the papers [21], struction to globally piecing together solutions to [22] by Spencer which systematically applied sheaf a problem that has local solutions. This frame- cohomological methods to issues related to divi- work introduces the higher cohomology groups sors and arising from classical algebraic geometry. q J J H (X, ) of a sheaf on a space X. For X an Essentially, they introduced the exponential sheaf J n-dimensional compact complex manifold and sequence the sheaf associated to a holomorphic vector bun- exp 0 → Z →O →O∗ → 1 dle, the duality theorem of Kodaira and Serre is X X √ q J ∗  n−q n J∗ (exp f = e2π −1f) and drew conclusions from its (1) H (X, ) H (X,ΩX ( )). exact cohomology sequence when the identification For X a compact Riemann surface, taking q = 0 and   ⊗ 1 2 J= Ω to be the sheaf of quadratic differen- ∗ divisor class X H1(O )  tials, (1) gives the natural isomorphism X group on X   dual space to the is made. Particularly noteworthy was their proof of (2)  quadratic differentials   H1(Θ ), X the Lefschetz (1, 1) theorem on X  fundamental classes where ΘX is the sheaf of holomorphic vector fields  Hg1(X) on X. This observation was to provide a key hint of divisors about how to measure variation of complex struc- ture in higher dimensions. where From several complex variables, important were Hg1(X) = H1,1(X,C) ∩ H2(X,Z) results such as the direct-image theorem of Grauert and, especially, the Cauchy-Riemann operator ∂ . A is the Hodge group of integral (1, 1) classes. I complex manifold may be given either by a coor- remember Don relating to me how he and Kodaira excitedly presented their proof to Lefschetz, who dinate covering {Uα,zα} with complex analytic glueing data grumpily remarked that, first, he couldn’t under-  stand the fancy sheaf theory and, second, in any z = f (z )inU ∩ U (3) α αβ β α β case he had proved the theorem. (The latter needs = ∩ ∩ fαβ(fβγ (zγ )) fαγ (zγ )inUα Uβ Uγ a caveat in that Lefschetz’s argument for the semi- simplicity of monodromy acting in the homology or by giving the ∂ -operator of a Lefschetz pencil was incomplete—only with ∂ : A0(X) → A0,1(X) Hodge’s work was the result established. To this day all proofs of the (1, 1) theorem require Hodge satisfying the integrability conditions theory.) 2 (4) ∂ = 0. Deformation theory had been on Spencer’s mind for some time prior to his work with Kodaira (cf. Here, we are given an almost complex structure [18], [17], [20]). As he explained it to me, the issue  J : T (X) → T (X) was that they did not know what should play the role in higher dimension of quadratic differentials J2 =−Id in the Teichmüller theory on Riemann surfaces. The and Ap,q(X) denotes the space of global smooth dif- breakthrough came with (2). With that major “hint” ferential forms of type (p, q) relative to J. The everything began to fall into place, leading to the equivalence of the definitions amounts to proving papers [11], [5], [10], [12] (and relatedly [13], [14], that (X,J) satisfying (4) has a covering by holo- [23]), which brought deformation theory into the morphic coordinate charts; in effect it must be core of complex algebraic geometry. shown that there are enough local solutions to Before describing their theory, I want to say that deformation theory seems to some extent to have = ∂f 0. “been in the air”. In particular, the important work This was proved in the real-analytic case by Eckmann- by Frölicher-Nijenhuis independently established Frölicher and in the C∞ case by Newlander-Nirenberg. the rigidity theorem stated below, and their papers These two approaches to complex manifolds are on the calculus of vector-valued differential forms reflected respectively by the Cˇech and Dolbeault influenced the work of Kodaira-Spencer as well as methods of representing sheaf cohomology; they will that of many others working in related areas.

JANUARY 2004 NOTICES OF THE AMS 21 Turning to the Kodaira-Spencer deformation Thus around a point of Xt0 the deformation is triv- theory, intuitively a deformation of a compact com- ial; i.e., is biholomorphic to a product. We may plex manifold X is given by a family {Xt }t∈B of then expect that the global nontriviality is measured compact complex manifolds containing X = X as t0 cohomologically. In overlaps we will have (cf. (3)) a member. In case X is a smooth complex projec- (i) zα = fαβ(zβ,t) tive algebraic variety given by homogeneous poly- (7) (ii) fαβ(fβγ (zγ ,t),t) = fαγ (zγ ,t). nomial equations If we think of the complex manifold as being de- X ={P (x ,...,x ) = 0}, λ 0 N scribed by the glueing data (7)(i), the vector fields

one may imagine deforming X by varying the poly- i ∂fαβ(zβ,t) ∂ nomials to Pλ(x0,...,xN ,t), with Pλ(x0,...,xN , 0) = (8) θ (t) =: αβ ∂t ∂zi Pλ(x0,...,xN ), and setting α

Xt ={Pλ(x0,...,xN ,t) = 0}, describe the variation of the glueing data and hence the variation of the complex structure. Here, provided that Xt remains of constant dimension for ∈ 1 { } ˇ θαβ(t) C ( Uα(t) , ΘXt ) is a Cech cochain for the |t| <$(in which case it will be smooth). More for- tangent sheaf ΘXt relative to the open covering mally, a deformation of X is given by Uα(t) = Uα ∩ Xt of Xt. Differentiation of (7)(ii) with X respect to t shows that the Cˇech coboundary (5) ↓ π δ{θαβ(t)}=0, B X and hence there is defined a cohomology class where and B are complex manifolds and π is a   smooth (i.e., the differential π∗ has maximum rank) { }∈ 1 θαβ(t) H ΘXt proper holomorphic mapping. The fibres −1 Xt = π (t) of (5) are then compact, complex man- which gives the Kodaira-Spencer mapping (6). A  ifolds, and we assume that Xt0 X for a reference basic result of the theory is point t ∈ B. The Kodaira-Spencer theory is local, 0 The family (5) is locally trivial if, and only if, and so we may think of B as an open neighborhood (9) all ρ are zero. of the origin in Cm. For the most part, under- t standing the case m = 1 will suffice, and so unless Behind this result are a number of subtleties: to stated otherwise we will restrict to this situation show that ρt is well defined, to understand the sit- and denote by t a coordinate on B. = = = uation when ρt0 0 but ρt 0 for t t0 (jumping Kodaira and Spencer sought to understand as of structure), etc. fully as possible the properties of local deforma- The other way of describing the family (5) may tions (5). They asked questions such as be expressed as follows. First, the Kodaira-Spencer 1. When do deformations (5) exist? mappings (6) give, in a precise sense, the obstruc- 2. What properties of X are stable under defor- tion to lifting ∂/∂t locally over the base to a mation? holomorphic vector field on X. If ∂/∂t lifts to a holo- q J 3. How do the cohomology groups H (Xt , t ) morphic vector field v with π∗(v) = ∂/∂t, then the behave in a family (5)? holomorphic flow of v gives biholomorphisms Here, we may think of J as the sheaf on X associ- ∼ → ated to a holomorphic vector bundle there, and then (10) Xt0 Xt Jt denotes the restriction of J to Xt. To study these questions they used both of the and therefore a trivialization ways discussed above of describing a complex XX × B. manifold. We shall use the first local coordinate t0 method to define the basic invariant of (5), the Now, as may be seen using a partition of unity, there Kodaira-Spencer mappings is no obstruction to lifting ∂/∂t as a C∞ vector   field. The resulting diffeomorphisms (10) may be → 1 (6) ρt : Tt B H ΘXt . used to transport the complex structures on Xt back to X . Analyzing this carefully, on X we may Intuitively, these measure the first-order variation t0 of the complex structure. Since the projection express the Cauchy-Riemann operator ∂t for Xt as mapping π in (5) has maximal rank, shrinking ∂t = ∂ + θ(t), B if necessary we may cover X by coordinate neighborhoods Uα with local coordinates (zα,t) where θ(t) is a ΘX-valued (0, 1) form having the local such that expression = j ⊗ i π(zα,t) = t. θ(t) θ(t) i ∂/∂zj dz¯ ,

22 NOTICES OF THE AMS VOLUME 51, NUMBER 1 where The result (13) is also a consequence of Grauert’s 2 direct-image theorem. θ(t) = θ1t + θ2t /2 +··· Some of the deepest results in the theory come is a convergent series in t. The integrability con- by using nonlinear elliptic theory. The joint paper [5] dition with Louis Nirenberg was in response to the ques- 2 = tion: Given a class {θ }∈H1(Θ ) , does there exist a ∂t 0 1 X family (5) with gives a series of relations ={ }  ρt0 (∂/∂t) θ1 ?  =  ∂θ1 0  From the discussion above, we see that what is ∂θ + 1 [θ ,θ ] = 0 (11)  2 2 1 1 required is to construct a convergent series   . = + 2 +··· . θ(t) θ1t θ2t /2 such that (11) is satisfied. The first equation is The first one implies that θ defines a Dolbeault 1 automatic, but the second and higher ones present cohomology class cohomological obstructions: e.g., if {θ }∈H0,1(Θ ). 1 ∂ X 2 {[θ1,θ1]}=0inH (ΘX ), The second implies that the Dolbeault cohomology then we cannot find θ2 satisfying the second equa- class tion in (11), and so forth. The main result in [5] is {[θ ,θ ]}∈H0,2(Θ ) 1 1 ∂ X 2 (15) If H (ΘX ) = 0, then there exists a family (5) such that the Kodaira-Spencer mapping defined by the bracket [θ1,θ1] should be zero. Collectively the equations (11) are equivalent to → 1 ρt0 : Tt0 B H (ΘX ) is an isomorphism. ∂t θ(t) = 0 These methods were extended by Kuranishi to so that   { }∈ 0,1 prove the existence of a local universal family (5) θ(t) H ΘXt . ∂t where B is an analytic subvariety in an open neigh- 1 The relation between the two approaches is borhood of the origin in H (ΘX ) with 1 2 expressed by the result dim B ≥ dim H (ΘX ) − dim H (ΘX ).

(12) Under the Dolbeault isomorphism Another stability result from Kodaira and 1 0,1 Spencer’s last paper [12] on deformations of H (ΘX )  H (ΘX ) t ∂t t complex structure is one has (16) If X is a Kähler manifold, then the Xt are ρt (∂/∂t) ={θ(t)}. also Kähler for t close to t0. This shows quite clearly how the Kodaira-Spencer As shown later by Hironaka, this result is false classes give the variation of complex structure. globally: the limit of Kähler manifolds may not be The second method is more amenable to using Kähler. the methods of linear elliptic partial differential Before turning to some other aspects of equations for deformation theory. For example, Kodaira-Spencer deformation theory and the work using the continuity of the eigenvalues in the dis- it stimulated, I would like to mention one particu- crete spectrum of an elliptic operator, Kodaira and lar aspect of the one-variable work of Schiffer and Spencer showed Spencer that has had significant impact in current algebraic geometry. Namely, the projectivized tan- (13) (Upper semi-continuity): For t close to t0, gent space PH1(Θ ) to the local deformation of a dim Hq(X , J ) ≤ dim Hq(X , J ). X t t t0 t0 smooth algebraic curve is, in a natural way, the q−1 J = q+1 J = If H (Xt , t0 ) H (Xt , t0 ) 0, then image space for the bicanonical mapping equality holds. 1 (17) ϕ2K : X → PH (ΘX ), As a corollary of this together with (9), they deduced the following result, which was proved indepen- which one may think of as using a basis for the qua- dratic differentials as homogeneous coordinates. dently by Frölicher and Nijenhuis. Mathematically, if for a point p ∈ X we let ΘX (p) 1 (14) If H (X,ΘX ) = 0, then the family (5) is locally denote the sheaf of vector fields with a first order trivial. pole at p, then there is an exact sheaf sequence

JANUARY 2004 NOTICES OF THE AMS 23 → → → → Hodge structure, which has had rich applications (18) 0 ΘX ΘX (p) ΘX (p) p 0 . in algebraic geometry. Moreover, the Kodaira- In terms of a local coordinate z centered at p, Spencer framework quickly spawned a number of sections of ΘX (p) are deformation theories of other structures, including a discrete subgroups of semisimple Lie groups θ = + b + b z +··· ∂/∂z z 0 1 (Calabi, Weil, and Matsushima, among others), and deformation theory of algebras (Gerstenhaber).  C and ΘX (p) p is a skyscraper sheaf supported Over the years these theories have been refined, at p, and the right-hand mapping in (18) is expanded, extended, and applied. (I shall not θ a. attempt any summary.) In the setting of general algebraic geometry, the The bicanonical mapping (17) then sends p to the Kodaira-Spencer theory became quickly absorbed, 1 line in H (ΘX ) spanned by adapted, and greatly extended by Grothendieck and his school. Together with the complementary a ∂ (19) δ ,a= 0, study of global moduli, most especially of marked z ∂z algebraic curves initiated by , one sees that some nearly fifty years after its inception, where δ is the coboundary map deformation theory, both local and global, is →δ 1 absolutely central in modern algebraic geometry ΘX (p) p H (ΘX ) and its interfaces with other areas, including string in the exact cohomology sequence of (18). By def- theory (quantum cohomology) and mirror sym- inition, ϕ2K (p) is the Schiffer variation associated metry (Calabi-Yau manifolds). to p. From (19) we see that References = P 1 ϕ2K (p) 0in H (ΘX (p)). [1] G. F. D. DUFF and D. C. SPENCER, Harmonic tensors on manifolds with boundary, Proc. Nat. Acad. Sci. U.S.A. As explained to me by Don, this has the following 37 (1951), 614–619. intuitive geometric meaning: To first order, the de- [2] ——— , Harmonic tensors on Riemannian manifolds formation with tangent ϕ2K (p) arises by leaving un- with boundary, Ann. of Math. 56 (1952), 128–156. changed the complex structure on X \{p} and [3] P. R. GARABEDIAN and D. C. SPENCER, Complex boundary changing it at p by a δ-function. This can all be value problems, Trans. Amer. Math. Soc. 73 (1952), made mathematically precise, and Schiffer varia- 223–242. tions have had a variety of important applications [4] V. K. A. M. GUGENHEIM and D. C. SPENCER, Chain homo- in contemporary algebraic geometry, including re- topy and the de Rham theory, Proc. Amer. Math. Soc. 7 (1956), 144–152. sults on the global moduli of Riemann surfaces and [5] K. KODAIRA, L. NIRENBERG, and D. C. SPENCER, On the on the higher Chow groups of algebraic cycles by existence of deformations of complex analytic Collino and others. structures, Ann. of Math. 68 (1958), 450–459. In addition to variation of complex structures, [6] K. KODAIRA and D. C. SPENCER, On the arithmetic gen- Kodaira and Spencer also considered deformations era of algebraic varieties, Proc. Nat. Acad. Sci. U.S.A. of holomorphic principal bundles [11] (and their 39 (1953), 641–649. associated vector bundles) and the relative case of [7] ——— , Groups of complex line bundles over compact deformations of a compact complex codimension- Kähler varieties, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), one submanifold within a fixed ambient complex 868–872. manifold [15], [10] (the higher codimension case [8] ——— , Divisor class groups on algebraic varieties, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 872–877. being done by Kodaira). In all these cases there is [9] ——— , On a theorem of Lefschetz and the lemma of a Kodaira-Spencer map encoding the first-order Enriques-Severi-Zariski, Proc. Nat. Acad. Sci. U.S.A. 39 variation, and results (including an existence (1953), 1273–1278. theorem) analogous to those given above were [10] ——— , A theorem of completeness for complex established. In [11] a rich set of examples illus- analytic fibre spaces, Acta Math. 100 (1958), 281–294. trating the general theory is worked out in detail. [11] ——— , On deformations of complex analytic Soon after the publication of the fundamental pa- structures. I, II, Ann. of Math. 67 (1958), 328–466. pers [11] there was a flood of results applying the [12] ——— , On deformations of complex analytic general theory. Among these were the rigidity of structures. III, Ann. of Math. 71 (1960), 43–76. simply connected homogeneous Kähler manifolds [13] ——— , On the variation of almost-complex struc- ture, Algebraic Geometry and Topology. A Symposium (Bott) and the rigidity of hyperbolic symmetric in Honor of S. Lefschetz, Princeton Univ. Press, 1957, compact complex manifolds having no unit disc pp. 139–150. factor in their universal coverings (Calabi-Vesentini). [14] ——— , Existence of complex structure on a differ- In the case of a compact Kähler manifold, a varia- entiable family of deformations of compact complex tion of complex structure leads to a variation of manifolds, Ann. of Math. 70 (1959), 145–166.

24 NOTICES OF THE AMS VOLUME 51, NUMBER 1 [15] ——— , A theorem of completeness of characteristic systems of complete continuous systems, Amer. J. Math. 81 (1959), 477–500. [16] J. J. KOHN and D. C. SPENCER, Complex Neumann problems, Ann. of Math. 66 (1957), 89–140. [17] A. C. SCHAEFFER and D. C. SPENCER, A variational method for simply connected domains, Construction and Applications of Conformal Maps. Proceedings of a Symposium, National Bureau of Standards, Appl. Math. Ser., No. 18, U.S. Government Printing Office, 1952, pp. 189–191. [18] M. SCHIFFER and D. C. SPENCER, A variational calculus for Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A. I. Math.-Phys. 1951, no. 93. [19] ——— , On the conformal mapping of one Riemann surface into another, Ann. Acad. Sci. Fenn. Ser. A. I. Math.-Phys. 1951, no. 94. Eugenio Calabi, Halsey Royden, Don Spencer, and Phillip [20] ——— , Some remarks on variational methods ap- Griffiths hiking near Palo Alto, CA, in the 1960s. plicable to multiply connected domains, Construction and Applications of Conformal Maps. Proceedings of Spencer [4] studied deformations of a class of a Symposium, National Bureau of Standards, Appl. G -structures which generalize foliations on a Math. Ser., No. 18, U.S. Government Printing Office, 1952, pp. 193–198. manifold. [21] D. C. SPENCER, Cohomology and the Riemann-Roch In his seminal paper [7], Spencer undertook a theorem, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 660–669. program to extend these results to structures [22] ——— , A generalization of a theorem of Hodge, Proc. corresponding to the pseudogroups of Lie and Élie Nat. Acad. Sci. U.S.A. 38 (1952), 533–534. Cartan. He had a complete vision about how to [23] ——— , A spectral resolution of complex structure, approach the theory of overdetermined systems International Symposium on Algebraic Topology, of partial differential equations through a whole Universidad Nacional Autónoma de México and series of new tools and to use them to study the UNESCO, Mexico City, 1958, pp. 68–76. theory of deformations of pseudogroup structures. Hubert Goldschmidt Most of it is described in [7]; it would take many years to fully understand, to develop and refine I first met Don Spencer when I was a sophomore everything which he formulated there, and to carry at Princeton University during the winter of out his program. Spencer’s ideas exerted a profound 1960–61. I first encountered his vision of mathe- influence on most mathematicians who worked in matics through his Advanced Calculus book [6] this field, in particular, on M. Kuranishi, R. Bott, written with H. Nickerson and N. Steenrod, which S. Sternberg, D. Quillen, V. Guillemin, B. Malgrange, had such an influence on a whole generation of A. Kumpera, Ngô Van Quê, and me. undergraduates at Princeton, and then through an The pseudogroups of Lie and Cartan arise as informal seminar held in his office. Don was to be transformation groups of geometric structures my thesis advisor for my senior thesis. His enthu- and are defined by systems of partial differential siasm and romantic view of mathematics, his equations. Spencer’s program consisted in first passion for the subject permeated his teachings: research in mathematics was something to be studying the geometric structures in terms of the enjoyed. His boundless energy was legendary. He associated systems of partial differential equa- also taught us to persevere in one’s vision, not to tions. Subsequently, it would lead to a deformation be led astray and not to deviate from one’s path. theory of structures on manifolds defined by I was fortunate to have been associated with Don pseudogroups which would incorporate all the as his student, colleague, and collaborator; to have fundamental mechanisms of the theory of defor- been led by him to the forefront of an exciting new mation of complex structures. In particular, an field at its very outset; and to have witnessed all extension of the Newlander-Nirenberg theorem to its beautiful developments over the years. elliptic pseudogroups would be required to obtain In 1960 Kodaira and Spencer began extending coordinates for the deformed structures and to their theory of deformation of complex structures prove the existence of local deformations. to other classes of geometric structures. First, in [3] The first step was carried out by studying Kodaira considered structures corresponding to the system of linear partial differential equations Élie Cartan’s primitive pseudogroups of biholo- associated to a pseudogroup in terms of Ehres- morphic transformations. Then Kodaira and mann’s theory of jets and constructing a resolution of the sheaf of vector fields which are the infini- Hubert Goldschmidt is a visiting professor at Columbia Uni- tesimal transformations of the pseudogroup. This versity. His email address is [email protected]. resolution was introduced in order to interpret

JANUARY 2004 NOTICES OF THE AMS 25 Ph.D. Students of Donald C. Spencer Arthur Grad (1948) infinitesimal deforma- of the Spencer sequence. Spencer asserted that this Louis N. Howard (1953) tions as elements of a cohomology is finite-dimensional and foresaw the Pierre E. Conner (1955) cohomology group. implications of the vanishing of certain groups. Mum- Robert Hermann (1956) Here, Spencer intro- ford pointed out to Spencer in 1961 that the dual of Joseph J. Kohn (1956) duced fundamental the δ-complex is a Koszul complex; this remark gave Bruce L. Reinhart (1956) tools for the theory of a direct proof of Spencer’s assertion. This led Guillemin Patrick X. Gallagher (1959) overdetermined systems and Sternberg to conjecture that Cartan’s notion of Alan L. Mayer (1961) of partial differential involutiveness, as reformulated by Matsushima, is Richard T. Bumby (1962) equations, which marks equivalent to the vanishing of the δ-cohomology. This Phillip A. Griffiths (1962) the beginning of a new equivalence was proved by Serre using commutative Michael E. Ash (1963) era in their study. His algebra and would provide the link between Spencer’s Joseph E. D’Atri (1964) approach—now called theory and Cartan’s methods. It is a remarkable W. Stephen Piper (1966) Spencer theory—was fact that Cartan’s notion of involutiveness can be William J. Sweeney (1966) more in the spirit of Lie, recast in terms of this cohomology. Techniques from Roger A. Horn (1967) with a full use of modern homological algebra and commutative algebra could Frank W. Owens (1967) concepts; the equations now provide tools for the study of overdetermined Barry MacKichan (1968) are studied directly, in systems and for new proofs of the celebrated Cartan- Laird E. Taylor (1968) contrast to the existing Kuranishi prolongation theorem. Sherman H. C. Hsiao (1972) Cartan-Kähler theory, 3. In view of proving the existence of solutions Robert J. V. Jackson (1972) where they are recast of real-analytic systems, Spencer formulated the Robert E. Knapp (1972) within the framework of so-called δ-estimate in terms of the δ-complex. Charles Rockland (1972) exterior differential sys- For a real-analytic system satisfying appropriate Constantin N. Kockinos (1973) tems. It was soon real- conditions, it would give the convergence of power Marshall W. Buck (1974) ized that this work was series solutions and thus the existence of local Suresh H. Moolgavkar (1975) not limited to equations solutions. The δ-estimate was subsequently proved John E. Lindley (1977) arising from pseudo- by Ehrenpreis, Guillemin, and Sternberg, and later Jack F. Conn (1978) groups, and a systematic by Sweeney. Malgrange realized that this estimate study of the formal the- is essentially equivalent to the “privileged neigh- ory of overdetermined borhood theorem” of Grauert and used this systems incorporating these methods was under- theorem together with the method of majorants taken. In fact, the methods introduced in [7] turned to prove a direct existence theorem for analytic out to provide most of the crucial elements for the systems (see [5]). study of overdetermined systems both from the for- These methods would give rise to the basic mal point of view and within the context of real- existence theorems for overdetermined systems analytic systems, and we now evoke several of of partial differential equations which guarantee the their aspects: existence of sufficiently many formal solutions for 1. To a system of linear equations or to a linear an arbitrary system, linear or nonlinear. In the case differential operator, Spencer associated complexes of real-analytic systems, the results mentioned of differential operators, the so-called naive and in (3) would then provide us with the existence of sophisticated Spencer sequences. Intrinsic con- local solutions. structions of these sequences were later given by In [8] Spencer formulated a generalization of the Bott, Quillen, and Goldschmidt. The cohomology of ∂ -Neumann problem for overdetermined elliptic these complexes, which is now called the Spencer systems on small convex domains in the hope of cohomology of the operator (or of the equations), proving the exactness of the sophisticated Spencer is a crucial ingredient in the study of overdeter- sequence associated to such equations. That this mined systems; in fact, it was later shown to be exactness holds in general for such systems is now equal to the cohomology arising from the com- called the Spencer conjecture. The Spencer- patibility conditions for the given differential Neumann problem was solved, and the exactness operator. Quillen also provided the proof of of the Spencer sequence was proved in several Spencer’s assertion made in [7]: if the system is special cases by Spencer’s students MacKichan, elliptic, the associated sophisticated Spencer Sweeney, and Rockland (see [1], Chapter X). By sequence is an elliptic complex. This result is in adapting Henri Cartan’s argument for the exactness fact critical for the deformation theory of structures of the Dolbeault sequence, in [7] Spencer also defined by elliptic pseudogroups. proved the exactness (in the C∞-sense) of the 2. Essential information about a system of differ- Spencer sequence under the assumption that the ential equations is contained in its δ-cohomology elliptic operator has real-analytic coefficients. (now also called Spencer cohomology) groups. In This last result implies that the Spencer se- fact, they are the cohomology groups of the so-called quence of an elliptic pseudogroup, whose equations δ-complex, which first appeared as a subcomplex have real-analytic coefficients, is exact and gives rise

26 NOTICES OF THE AMS VOLUME 51, NUMBER 1 to an interpretation of the infinitesimal deforma- tions as elements of a cohomology group, just as in the case of complex structures. Another aspect of the methods of [7] for studying deformation theory is the recasting of the notion of an almost- structure (corresponding to the pseudogroup) as a section of one of the vector bundles appearing in a nonlinear sequence corresponding to the naive or to the sophisticated Spencer sequence. The integrability condition for the almost-structure is then formulated in terms of this sequence. The second fundamental theorem for a pseudogroup asserts the existence of coordinates for an almost- structure satisfying this requisite integrability con- dition. For elliptic pseudogroups whose equations have real-analytic coefficients, the second funda- Spencer and De Rham, India, 1964. mental theorem was proved by Malgrange in [5]. The References proof of this generalization of the Newlander- Nirenberg theorem was inspired by the one which [1] R. BRYANT, S. S. CHERN, R. GARDNER, H. GOLDSCHMIDT, and P. GRIFFITHS, Exterior Differential Systems, Math. Sci. Spencer gave for the exactness (in the C∞-sense) of Res. Inst. Publ., vol. 18, Springer-Verlag, 1991. the sophisticated linear Spencer sequence and re- [2] H. GOLDSCHMIDT and D. C. SPENCER, On the non-linear quires all of the mechanism developed by Spencer. cohomology of Lie equations. I, II, Acta Math. 136 This completed Spencer’s program for studying (1976), 103–170, 171–239. deformations of structures initiated in [7] and [3] K. KODAIRA, On deformations of some complex pseudo- showed that indeed Spencer had introduced all group structures, Ann. of Math. 71 (1960), 224–302. [4] K. KODAIRA and D. C. SPENCER, Multifoliate structures, the essential components for the study of the de- Ann. of Math. 74 (1961), 52–100. formation theory of pseudogroup structures. [5] B. MALGRANGE, Equations de Lie. I, II, J. Differential In [5] Malgrange introduced the formalism of Geom. 6 (1972), 505–522; 7 (1972), 117–141. differential calculus à la Grothendieck into the [6] H. K. NICKERSON, D. C. SPENCER, and N. E. STEENROD, Ad- study of pseudogroups and reworked the Spencer vanced Calculus, Van Nostrand, Princeton, NJ, 1959. mechanism for studying deformations of struc- [7] D. C. SPENCER, Deformation of structures on manifolds ture. Also, he clarified the correspondence between defined by transitive, continuous pseudogroups. I, II, Ann. of Math. 76 (1962), 306–398, 399–445. the linear Lie equations and their finite forms. [8] ——— , Deformation of structures on manifolds defined Many of Spencer’s ideas concerning pseudogroups by transitive, continuous pseudogroups. III, Ann. of now appeared in a rigorous form. Malgrange’s work Math. 81 (1965), 389–450. was utilized by Goldschmidt and Spencer in their [9] ——— , Overdetermined systems of linear differential paper [2] to study the second fundamental theo- equations, Bull. Amer. Math. Soc. 75 (1969), 179–239. rem for an arbitrary pseudogroup. They introduced Enrico Bombieri the nonlinear Spencer cohomology of a transitive pseudogroup, whose vanishing implies the valid- The earliest mathematical works of Don Spencer, ity of the second fundamental theorem for the reflecting the influence of his advisor J. E. Little- pseudogroup. By means of a spectral sequence wood and Littlewood’s collaborator, G. H. Hardy, argument involving the nonlinear sequences, they were in analytic number theory. A particularly im- portant problem on which Spencer worked was to showed that this nonlinear Spencer cohomology show that a progression-free sequence of positive depends only on the Lie algebras of all formal integers cannot have positive density. The paper infinitesimal transformations of the pseudogroup. [1], written jointly with Raphaël Salem, disproved Many of the main ingredients of their solution of in a clever fashion a then widely held conjecture. the “integrability problem” for flat pseudogroups The particular problem has been of lasting inter- appear here. In fact, their results concerning est, with major progress by Szemerédi and quite Spencer cohomology allowed them to use recently by Gowers, who was awarded a Fields Guillemin’s Jordan-Hölder decompositions and Medal in part for his work on it. In addition to this Galois theory type methods to provide a proof of the second fundamental theorem for all transitive Enrico Bombieri is professor of mathematics at the Insti- pseudogroups on Euclidean space containing the tute for Advanced Study, Princeton. His email address is translations, the so-called flat pseudogroups. [email protected].

JANUARY 2004 NOTICES OF THE AMS 27 and the related paper [2], Spencer’s main other Paul Garabedian works in analytic number theory were in Dio- phantine approximations, again reflecting the in- Don Spencer started his career at the Massa- terest of the English school at that time. chusetts Institute of Technology doing research in the theory of functions of one complex variable. References During World War II he came under the influence [1] R. SALEM and D. C. SPENCER, The influence of gaps on of Courant at University and started to density of integers, Duke Math. J. 9 (1942), 855–872. apply variational methods to the coefficient prob- [2] ——— , On sets of integers which contain no three lem for univalent functions. Max Schiffer had al- terms in arithmetical progression, Proc. Nat. Acad. ready initiated similar studies in Palestine that Sci. U.S.A. 28 (1942), 561–563. were to change the whole climate in this field. After the war they combined forces both at Stanford and Bohous Cenkl at Princeton and made significant contributions to our knowledge about Riemann surfaces. Don re- We always called him Don. He disliked formal- cruited Arthur Grad to write a Ph.D. thesis at Stan- ities and despised political intrigue. As practically ford on univalent functions, and Grad went on to the only visitor permitted from Czechoslovakia to guide early efforts in support of applied mathe- Stanford in the 1960s, I had the good fortune to matics at the Office of Naval Research and at the first meet him there. That was the time that the National Science Foundation. Stanford Linear Accelerator (SLAC) was being built. Al Schaeffer joined Don in a study of the dif- Don’s primary research focus was the study of ferential equations for extremal functions in con- overdetermined linear systems, but his passion formal mapping that contributed significantly to for theoretical physics was one of many side in- steady progress on the Bieberbach conjecture that terests that he shared with Ko- ended with the complete solution of the problem.1 daira. Expanding on Don and Ko- Their work was recognized by an award of the daira’s seminal work on Bôcher Prize from the AMS. I was fortunate to be deformations of complex struc- collaborating with Don during the period when he tures, Hermann was running a turned his attention from one variable to focus on seminar on deformations in the major questions about complex manifolds. His physics department. During that year, Don and I spent endless generous and vigorous encouragement of his hours discussing all types of prob- graduate students and younger colleagues led him lems beyond just mathematics and to exert a strong influence on the direction of physics. Don was always pushing mathematical research in this country during the me to learn things that I felt were second half of the last century. well beyond my capabilities. Louis Nirenberg There was a constant stream of visitors in Don’s office—Borel, Hör- I think I first met Don Spencer when I was a grad- mander, Hermann, Kohn, Niren- uate student or young instructor. He came to visit berg, Sternberg, and many others. Courant occasionally and would sometimes be in- Spencer in California, Don made every one of us feel spe- vited to Courant’s home in New Rochelle during a around 1965. cial. He never singled anyone out, weekend, as I was. Later we joked about having but rather devoted his endless en- to help take care of Courant’s garden. ergy to those with no Ivy League education and to We became friends, and it was always a great future Nobel Prize winners alike. He never suc- pleasure to meet with Don in Stanford, Princeton, cumbed to any attempt by some to use his name or wherever. His love and enthusiasm for mathe- for self-promotion. matics were contagious. I heard him speak many After a heavy dose of mathematics, Don always times, and though I usually had trouble following, planned various outdoor activities for us, like a I came away with a positive glow—I liked the hike above Palo Alto on the ranch of his friend Jim melody. Rapley, a retired commander in the U.S. Navy. I have Don’s early work was on analytic functions of fond memories of numerous trips on horseback to one variable, but later he expanded to complex the Sierras with Don and his family. I greatly respected him and his life philosophy. I am personally grateful Paul Garabedian is professor of mathematics at the for the opportunity to have had Don as a teacher and Courant Institute of . His email address a very close friend to me and my family. is [email protected]. 1By Louis de Branges in 1985. Bohous Cenkl is professor of mathematics at Louis Nirenberg is professor emeritus at the Courant Northeastern University. His email address is Institute of New York University. His email address is [email protected]. [email protected].

28 NOTICES OF THE AMS VOLUME 51, NUMBER 1 manifolds. He wrote a series of famous papers with Kodaira on deformation of complex struc- ture. I visited the Institute for Advanced Study dur- ing the spring of 1958 and had the good fortune to write a joint paper with them. When Don tried to describe the problem to me, he kept putting it in extremely general terms. Finally I asked if he could write down a simple case. He had trouble doing this, but then Kodaira stepped to the black- board and without saying a word, wrote down a few simple equations. Later, after the work was done, I heard Don speak on it at his and Kodaira’s “Noth- ing Seminar”. I had trouble following the talk, for Don liked to use very abstract and most general con- cepts. About the “Nothing Seminar”: it was a seminar they ran which was completely informal, no planned program. People would talk about what they were working on, thinking about, or were stuck on. It was usually held around lunchtime and was indeed very informal. I remember once a participant had brought in his lunch, hard-boiled eggs, whose shells he scattered on the floor while Don lectured. Don didn’t raise an eyebrow. In 1963 we participated in a joint Soviet-American conference on partial differential equations in Novosibirsk. There were about two dozen American and one hundred twenty-five Soviet mathemati- cians. Toward the end of the conference the Soviet guides and translators, mainly women, voted Don the most handsome participant. Incidentally on his way to the Soviet Union, Don stopped in Cambridge, England; he had received his Ph.D. there some decades earlier. On his arrival he went to a bank to change money. The clerk looked at him and said, “Nice to see you again, Dr. Spencer.” On that visit to England he bought an umbrella, one of those long, rolled, elegant umbrellas we know from movies. Throughout the Novosibirsk confer- ence I kept admiring it. At the end he simply presented it to me. I treasured it for years. Don had a wonderful, open personality. He was the same natural person with everyone. He was extremely generous in giving of himself, offering warmth, friendship, and attention to everyone, concerned about their problems. I consider him the most considerate person I have ever met—a great gentleman. Once, at a dinner party, people were talking about Don, praising him, etc. At the end, my wife said, “Everyone loves Don, but I like him anyway.” I found it difficult to keep up with his later work. After he moved to Durango, Colorado, we didn’t meet very often, but as always, each meeting was a great pleasure. Once my wife and I visited him in Durango. He took us on a tour—to various bars and taverns. He was warmly welcomed in every one.

JANUARY 2004 NOTICES OF THE AMS 29