Ad Honorem Charles Fefferman

Communicated by Stephen Kennedy and Steven J. Miller

Charles Fefferman circa 2005.

1254 Notices of the AMS Volume 64, Number 11 Antonio Córdoba

Introduction The prestigious Wolf Prize of 2017 has been awarded to Charles Fefferman, ex aequo with Richard Schoen. Charles Fefferman (Charlie) is a mathematician of the first rank whose outstanding findings, both classical and revolutionary, have inspired further research by many others. He is one of the most accomplished and versatile mathematicians of all time, having so far contributed with fundamental results to harmonic analysis, linear PDEs, several com- plex variables, conformal geometry, “Problems quantum mechanics, fluid mechan- ics, and Whitney’s theory, together seem to with more sporadic incursions into select me!” other subjects such as neural net- works, financial mathematics, and crystallography. I have requested the help of a distinguished group of his friends and collaborators to provide reflections on Fefferman’s contributions to their respective fields. Before reading their remarks, it will be interesting to hear from Charlie himself: “Problems seem to select me! It’s just so exciting. A problem sort of chooses you, and you can’t stop thinking about it. At first, you try something, and it doesn’t work. You get clobbered! You try something else and get clobbered again! Eventually you get some insights and things begin to come together. Everything starts to move. Everyday things can look different. It’s very exciting. Eventually you manage to solve it all, and that’s a great feeling!” Born on April 18, 1949, Fefferman was a child prodigy Charlie with daughters Nina (in red) and Lainie who at the age of seventeen graduated from the University around 1983. of Maryland, where he received a joint bachelor’s degree in mathematics and physics. In 1969 he gained his PhD at Princeton University under the supervision of Eli Stein. tied, but painting was never in the lead. Eventually he In 1971, at the University of Chicago, he became the realized that he was much better at math. After knowing youngest full professor at any US college or university, a him for thirty-five years, if I had to account for his choice, fact that merited his appearance in Time and Newsweek I’d say that his passion for the beauty of math was what magazines in that same year. overtook everything else. I think that for him it’s almost an Charlie returned to Princeton University in the fall of addiction to the art of beautiful mathematics…Whenever 1974, where since then he has pursued his mathematical I ask him to try to explain his work to me, his eyes sparkle career. In 1975 he and his wife, Julie, got married and and his voice and gestures are infused with an animation went on to have two daughters, Nina and Lainie. Nina that is not present at any other time.” is a computational biologist who applies mathematical I met Charlie at the University of Chicago during models to complex biological systems, while Lainie is a the academic year 1971–1972. He was then a recently composer and holds a PhD in musical composition from appointed full professor and I was a first-year graduate Princeton University. Charlie has a brother, Robert, who student. The Calderón-Zygmund seminar was probably is also a mathematician and professor at the University of the place where we first got acquainted. At the end of Chicago. Julie has this to say about her husband: “When that academic year Charlie agreed to be my thesis advisor. Charlie was young, he fell in love: (1) with painting and art Let me add that for me it was a fantastic experience; we and (2) with math. He says that for a while the two were are of the same age, and at that time we became close Antonio Córdoba is the director of the Mathematics Institute friends. We played ping-pong together in the Eckhard Hall (ICMAT) and professor of mathematics at the Autonomous Univer- basement and had long conversations about science, art, sity of Madrid. His email address is [email protected]. movies, music, politics, and, of course, mathematics. For permission to reprint this article, please contact: I had the privilege of being his first graduate student, [email protected]. thereby initiating a set that now contains more than DOI: http://dx.doi.org/10.1090/noti1606 twenty elements. There is no doubt in my mind that the

December 2017 Notices of the AMS 1255 nontangential version, the Lusin area integral 푆, for which it was known that the mappings 푔 ↦ 푔(푓), 푔 ↦ 푆(푓) were of weak-type 1 and bounded on 퐿푝, 1 < 푝 < ∞. Another more arcane (“tangential”) variant arose in some problems, ∗ the function 푔휆 , defined by (1) 푡 푛휆 푔∗(푓)(푥)2 = ∫ |∇푢(푥 − 푦, 푡)|2 ( ) 푡1−푛푑푦푑푡. 휆 푛+1 ℝ+ |푦| + 푡 Here 푢(푥, 푡) is the Poisson integral of a function 푓 on 푛 ∗ ℝ . In fact 푔휆 dominated both 푆 and 푔, and it could be ∗ 푝 proved that 푓 → 푔휆 (푓) was bounded on 퐿 if 1 < 휆 < 2 and 푝 > 2/휆; moreover, this failed when 푝 = 2/휆. It ∗ seemed that the right assertion was that 푓 ↦ 푔휆 (푓) was of weak-type 푝 if 푝 > 1, but I had no real idea how to proceed, and this clearly required a new approach. Charlie Fefferman with the ICMAT fluids team, In just a short few weeks Charlie came back with Madrid circa 2011, from left to right: Javier the proof. I was surprised not only by the speed with Gómez-Serrano, Angel Castro, Charles Fefferman, which he had accomplished it but by the strength of his Diego Córdoba, Francisco Gancedo mathematics. I will not describe his idea of the proof but only say that it provided him a guide to solving the next problem—one that I proposed to him soon thereafter. opportunity of enjoying Charlie’s advice and friendship This problem concerned highly oscillating singular is an experience we all will treasure. integrals. An example was the transformation 푇 ∶ 푓 → 푓∗퐾 where the kernel 퐾 (a distribution) is given by Elias Stein 푖 푒 |푥| (2) 퐾(푥) = , when 0 < |푥| ≤ 1 푛 Fefferman’s Early Work: The Epic Years (1969– |푥| 1974) and vanishes for |푥| > 1. It was known that the resulting 푝 I want to write about the first five years of Charlie 푇 was bounded on 퐿 , 1 < 푝 < ∞, but the hoped-for Fefferman’s major work, a brief period in which his many weak-type 1 result seemed out of reach. innovations transformed our views of several subjects in Charlie set himself to work on this, and again, within an intensive series of achievements unique in the history a short few weeks he proved the desired assertion. His of modern mathematics. ideas were as follows. To begin with he found the right restatement of the The Dissertation problem in its general form. It concerned a distribution kernel 퐾 of compact support (represented by a function I first met Charlie in the fall of 1967. He had started as a 퐾(푥), when 푥 ≠ 0) which for a fixed parameter 휃, with graduate student at Princeton the year before, but I was on leave that year. Our first real contact was after he had 0 ≤ 휃 < 1, satisfied the following two conditions: −휃 푛 whizzed through his qualifying exam and said he wanted (i) 퐾(휉)̂ = 푂(|휉| 2 ) as |휉| → ∞, to write his thesis with me. (ii) ∫|푥|≳|푦|1−휃 |퐾(푥 − 푦) − 퐾(푥)|푑푥 ≤ 퐴. At that time the “Calderón–Zygmund paradigm,” as it The standard Calderón–Zygmund kernels correspond to was later to be called, had already proved very successful the case 휃 = 0, while the highly oscillatory ones corre- in freeing parts of harmonic analysis from its heavy 1 spond to 0 < 휃 < 1, with 휃 = 2 for the particular case (2) reliance on complex methods and leading it to the open above. vistas in higher dimensions, with new challenges and With this incisive first step, he next decomposed an possibilities. arbitrary 푓 in 퐿1, for fixed 훼 > 0, in the standard way, 2 One question that particularly fascinated me was that writing 푓 = 푔 + Σ 푏 , with 푔 in 퐿 , ‖푔‖ 2 ≤ 훼, and the 푏 푝 푗 푗 퐿 푗 of developing further methods for proving 퐿 estimates, in supported on disjoint cubes, while particular where the critical limiting exponent for 푝 might 1 be greater than 1, unlike the standard singular integrals. A ∫ |푏푗|푑푥 ≈ 훼 and ∫ 푏푗푑푥 = 0. problem of this kind arose in the Littlewood–Paley theory |푄푗| 푄푗 푄푗 of square functions, and I proposed it to Charlie, eager to So 푇(푔) could be disposed with by the 퐿2 theory. Coming see what this promising eighteen-year-old could do. to the bad part, we let 퐵푗 be the ball surrounding 푄푗 To explain what was involved, there were first the (having the same center but twice the diameter). Here the more standard square functions, the 푔-function, and its ∗ main idea was to introduce the balls 퐵푗 , with the same center, but with Elias Stein is emeritus professor of mathematics at Princeton ∗ 1−휃 University. His email address is [email protected]. diam 퐵푗 ≈ (diam 퐵푗) .

1256 Notices of the AMS Volume 64, Number 11 훿 In 1936 S. Bochner had introduced the operators 퐵푅 defined by

2 (1 − |휉| )훿푓(휉)̂ for |휉| < 푅, 퐵훿(푓)(휉)̂ = { 푅2 푅 0 for |휉| ≥ 푅. These were later dubbed the Bochner–Riesz means (of order 훿). For 푛 = 1, 훿 = 0, these were the classical partial-sum operators that were closely related to the Hilbert transform, and thus one had 퐿푝-norm control for 1 < 푝 < ∞. Bochner had pointed out that for 푛 > 1, 푛−1 the order 훿 = 2 was the critical index in the sense 푛−1 that if 훿 > 2 , then 퐵푅(푓) = 푓 ∗ 퐾푅, where the kernels 퐾푅 are good approximations to the identity, while for 푛−1 훿 ≤ 2 things are quite different and depend critically on the oscillatory nature of the 퐾푅. By 1960 it was known that for each 푝, 1 < 푝 < ∞, one had 퐿푝 control for 푛−1 some 훿 = 훿(푝) < 2 (also convergence a.e. as 푅 → ∞), but these conclusions were far from optimal. In fact, 훿 one was led to expect that the 퐵푅 when 훿 = 0 were 푝 2푛 2푛 bounded in 퐿 for 푛+1 < 푝 < 푛−1 (in two dimensions 4 3 < 푝 < 4), together with the wider expectation that the 훿 1 1 1 퐵푅 were bounded outside that range when 훿 > 푛| 푝 − 2 |− 2 . Charlie’s challenge was to go decisively beyond what was then known in a quest to achieve these ultimate goals. In urging him to undertake this clearly difficult and uncertain effort, I could not be of much help, except for one thing I had observed the year before: when 푛 > 1 and 푝 is sufficiently close to 1, the Fourier transform of an 퐿푝 function can be restricted to the unit sphere 푆푛−1 of ℝ푛. More precisely, one had, in the present day terminology, the (퐿푝, 퐿푞) restriction phenomenon

Charlie circa 1976 at Princeton. 1 푞 푞 (3) (∫ |푓(휉)|̂ 푑휎(휉)) ≲ ‖푓‖퐿푝(ℝ푛) 푆푛−1 when 푞 = 2, for 푝 close to 1. The obvious suggestion (Only balls with diam 퐵푗 ≤ 1 are significant here.) The ∗ was to explore the possible implications of this to the BR contribution of 푇(푏푗) outside 퐵푗 can be handled by (ii), means. as in the standard situation. ∗ Here Charlie succeeded marvelously with a remarkable There remain the critical contribution of 푇(푏푗) in 퐵푗 /퐵푗 conclusion: whenever the (퐿푝, 퐿2) restriction phenomenon and the corresponding estimates of 푇(푏)̃ = ∑ 푇(푏 ̃ ) 훿 푗 held, the optimal bounds for 퐵푅 with the same 푝 would (where 푏푗̃ is a cleverly chosen replacement of 푏푗). At follow as a consequence. this stage he had prepared everything to go back to After this accomplishment, we had some brief discus- 퐿2 estimates, and he succeeded because he was able to sions on trying to extend the restriction phenomenon. By combine directly two facts in this setting, namely, dualizing the problem we succeeded in obtaining the opti- 푝 푞 −푛 휃 mal conclusion in two dimensions: the (퐿 , 퐿 ) restriction ‖푇(푏)‖̃ 퐿2 ≲ ‖(1 − Δ) 4 푏‖̃ 퐿2 and 4 4 ′ holds for 푝 = 3 − 휖, and 푞 = 3 + 휖 , more precisely for ′ 4 −푛 휃 3푞 = 푝 , with 1 ≤ 푝 < . All the above results are in ‖(1 − Δ) 4 푏‖̃ 2 ≲ 훼‖푏‖ 1 . 3 퐿 퐿 Charlie’s dissertation, published in 1970. Shortly thereafter, two exciting developments followed. The result in its elegance and power, achieved so First, L. Carleson and P. Sjölin showed that in two quickly, amazed me, as it would have anyone. Later his dimensions, whenever 훿 > 0, one had boundedness of approach became a subject of much wider interest, as 4 BR means for 3 ≤ 푝 ≤ 4. This they did in part by it was adapted to various other problems by S. Chanillo, adapting the philosophy for the restriction theorem in M. Christ, Rubio de Francia, and others. two dimensions. However gratifying this result was, it Because of these two striking successes, I had little left open the pressing question: how to obtain the 퐿푝 hesitation in urging Charlie to attack a major problem, boundedness for the BR means of order 훿 = 0 in the 4 one which had concerned me for a number of years—that range 3 < 푝 < 4. In other words, how to deal with the disc of the Bochner–Riesz means. To give some background: multiplier.

December 2017 Notices of the AMS 1257 ℝ, it consisted of those functions 퐹 on ℝ that arose as boundary values of the holomorphic functions 퐹 in 2 푝 2 ℝ+ that are uniformly bounded in 퐿 on all lines in ℝ+ parallel to ℝ. Formally, each such 퐹 can be represented as 퐹 = 푓 + 푖퐻(푓), where 푓 is real-valued and 퐻 is the Hilbert transform. Thus if 1 < 푝 < ∞, classical 퐻푝 was essentially real 퐿푝(ℝ). However, when 푝 ≤ 1, things were quite different, and 퐻푝 had all sorts of interesting properties either not valid for 퐿1 or totally missing for the trivial space 퐿푝 when 푝 < 1. By 1960 it was clearly time to see whether there was anything like real Hardy spaces in ℝ푛, for 푛 > 1. Here G. Weiss and I had some initial success. We defined the space 퐻1(ℝ푛) to consist of those 퐿1 functions for which, in addition, their Riesz transforms, 푅푗(푓), 푗 = 1, … , 푛, also belonged to 퐿1. It turned out that this space had a number of interesting features going beyond 퐿1, in particular a maximal characterization involving Poisson integrals in terms of 퐿1 (and not merely weak-퐿1). A further hint that 퐻1(ℝ푛) was the appropriate substitute for 퐿1(ℝ푢) came several years later, when it was seen that classical singular integrals, broadly speaking, took 퐻1(ℝ푛) to 퐿1(ℝ푛) and in fact preserved 퐻1(ℝ푛). Let us come to the other branch of the tree: BMO. That space (functions of bounded mean oscillation) appeared first in 1961 in the work of F. John and L. Nirenberg. A function 푓 on ℝ푛 was said to be in BMO if Eli Stein (center) with the Fefferman brothers at the 1 (4) sup ∫ |푓 − 푓푄|푑푥 ≤ 퐴 < ∞. conference to celebrate Stein’s sixtieth birthday at 푄 |푄| 푄 Princeton in 1991. Charlie is on the left, Robert on the 1 right. Here 푓푄 = |푄| ∫푄 푓푑푥, and the 푄 range over all cubes 푄 in ℝ푛. John and Nirenberg proved the remarkable fact that (4) Charlie’s answer was an unexpected shock. His remark- implies an analogous inequality with |푓 − 푓푄| replaced by 푝 푐|푓−푓푄| able counterexample showed that the hoped-for result |푓 − 푓푄| , for any 푝 < ∞, and in fact replaced by 푒 fails, except for the obvious case 푝 = 2. It dramatically for an appropriate positive 푐. transformed our view of the subject. This development This kind of result had immediate consequences: in and a number of Charlie’s other achievements are covered the work of John for rotation and strain of mappings and in Terry Tao’s section, so I will not say more about this in J. Moser’s elegant proof of the DiGiorgi–Nash estimates. direction of his work. Then in 1966 the relevance of BMO to harmonic analysis— and the possibility that it might serve as the appropriate The 퐻1-BMO Duality substitute for 퐿∞—became a little clearer when J. Peetre, In the fall of 1970, after spending his first postdoctoral S. Spanne, and I (quite independently) observed that the ∞ year in Princeton, Charlie took up a position at the standard singular integrals took 퐿 to BMO and, even University of Chicago. There he met Antoni Zygmund, better, that BMO was stable under these transformations. who almost immediately put a question to him: what is the With this we return to the question Zygmund asked Poisson integral characterization of BMO? This question Charlie: to characterize BMO by Poisson integrals. Char- led Charlie to think more about BMO and ultimately to lie’s answer was given in terms of three interrelated formulate and prove the famous aforementioned duality. assertions, each remarkable in its own right: (A charming reminiscence of his thinking about the (i) The dual space of 퐻1 is BMO. issues involved can be found in his contribution to the (ii) A function 푓 is in BMO if and only if collection All That Math: Portraits of Mathematicians as 푛 Young Readers.) ∞ 1 푓 = 푓0 + ∑ 푅푗(푓푗), where 푓0, 푓1, … , 푓푛 are all in 퐿 . Let me briefly recall some facts about 퐻 and BMO. The 푗=1 classical Hardy space 퐻푝, 푝 > 0, arose at the intersection of complex analysis and Fourier analysis. Rephrased in (iii) 푓 is in BMO if and only if 푡|∇푢(푥, 푡)|2푑푥푑푡 is a 2 the setting of the upper half-plane ℝ+ with its boundary Carleson measure.

1258 Notices of the AMS Volume 64, Number 11 The assertion (iii) was the answer Zygmund sought but equations, and his work involving the mapping theorem did not expect! Here 푢(푥, 푡), 푥 ∈ ℝ푛, 푡 > 0, is the Poisson and the Bergman kernel—to which I’ll limit myself. integral of 푓, and the statement (iii) was that While still at the University of Chicago, Charlie was 1 told by R. Narasimhan of an outstanding problem in sup ∫ 푡|∇푢(푥, 푡)|2푑푥푑푡 ≤ 퐴 < ∞, several complex variables: Suppose Ω1 and Ω2 are a pair 퐵 |퐵| 푇(퐵) of bounded domains in ℂ푛, 푛 > 1, each having a smooth where 푇(퐵) = {푥, 푡 ∶ |푥−푦| < 푟−푡, 0 < 푡 < 푟} is the “tent” (퐶∞) boundary and each being strongly pseudoconvex. over the ball 퐵 ⊂ ℝ푛, centered at 푦 of radius 푟, and 퐵 Suppose there is a bijective biholomorphism Φ from Ω1 ranges over all balls. It is noteworthy that measures of to Ω . Then does Φ extend to a smooth diffeomorphism this kind arose for 푛 = 1 in Carleson’s work in 1962 on 2 of Ω̄ to Ω̄ ? the corona conjecture. 1 2 It must be stressed that the case 푛 ≥ 2 differs essen- Charlie’s discovery reawakened interest in Hardy tially from the classical case 푛 = 1. First, there is no spaces and BMO, and a flurry of activity followed im- adequate theory of conformal mappings to help when mediately. Working together we developed a systematic 푛 ≥ 2. In particular, two theory of Hardy spaces, which in particular freed it from domains may be such that its reliance on harmonic functions. This showed that an Ω̄ and Ω̄ are close (in …as inspired element of 퐻푝 could be characterized in a variety of 1 2 퐶∞) without there existing different but equivalent ways: a biholomorphism between and natural in (i) in terms of maximal characterizations by general them. Moreover, pseudocon- conception as smooth approximations of the identity (not just vexity must enter the picture, restricted to Poisson integrals), one way or another. Af- it was difficult (ii) similarly, in terms of general square functions, and ter explaining the problem, (iii) in terms of Riesz transforms and their higher Narasimhan said to Charlie, and painful in analogues. “You must do that.” A consequence of this was the confirmation of the status Charlie was immediately execution… 1 1 푝 푝 of 퐻 as the rightful substitute for 퐿 (and 퐻 for 퐿 , 푝 < 1) taken with the problem. He in the theory of singular integrals; not only was it stable remembered a course he attended as a graduate student under the action of these operators, but appropriate on boundary behavior of holomorphic functions and hav- 1 maximal and square functions had 퐿 control, and there ing been intrigued by the Bergman kernel and resulting was also a useful Calderón–Zygmund decomposition in metric. The facts about these are as follows. this context. The Bergman kernel 퐾Ω(푧, 푤) of domain Ω is de- Another interesting by-product was the function 푓# termined by the fact that 푓 ↦ ∫Ω 퐾Ω(푧, 푤)푓(푤)푑푤 is defined by the orthogonal projection of 퐿2(Ω, 푑푤) onto the sub- 2 # 1 space of 퐿 holomorphic functions on Ω. A remarkable 푓 (푥) = sup ∫ |푓 − 푓푄|푑푥. 푥∈푄 |푄| 푄 feature was that if the Hermitian (Riemannian) metric 2 휕 푑푠 = ∑ (log 퐾Ω(푧, 푧))푑푗 ̄푧 , 푑푧푘 is attached to Ω, then In analogy with the John–Nirenberg theorem one had that 휕 푗̄푧 휕푧푘 푓# ∈ 퐿푝 implied that 푓 ∈ 퐿푝, if 푝 < ∞, and this was a biholomorphism Φ ∶ Ω1 → Ω2 induces an isometry in useful in resolving the problem of complex interpolation these metrics. between 퐻1 or BMO and 퐿푝, 1 < 푝 < ∞. Charlie’s idea was as inspired and natural in conception A significant further development started with Char- as it was difficult and painful in execution. In brief, what lie’s observation that the duality of 퐻1 and BMO can be had to be done was to carry out the following plan: after restated by saying each 푓 ∈ 퐻1 has an atomic decompo- fixing corresponding points 푃1 ∈ Ω1, 푃2 = Φ(푃1), follow sition. This meant that any such 푓 could be written as any geodesic on Ω1 starting at 푃1 and the corresponding geodesic Ω to infinite time when both reach the boundary ∑푗 휆푗푎푗, where each atom, 푎푗, was supported on a cube 2 1 at points 푄1 ∈ 휕Ω1 and 푄2 ∈ 휕Ω2. The mapping 푄1 ↦ 푄2 푄푗, with |푎푗(푥)| ≤ , ∫ 푎푗(푥)푑푥 = 0, and the constants |푄푗| should then yield the correspondence of the boundaries 휆 satisfied ∑ |휆 | < ∞. 푗 푗 푗 and thus give the 퐶∞ extension of Φ. In the hands of R. Coifman, C. Herz, R. Latter, G. Weiss, The first step therefore was analyzing the behavior of and others, the idea of atomic decomposition was greatly the Bergman metric, which required a clearer understand- developed, encompassing 퐻푝, 푝 < 1, and ultimately be- ing of the Bergman kernel, a question of great interest in coming the preferred starting point for various extensions its own right. The description of 퐾 he obtained was as of the theory. Ω follows. Assuming that Ω is a bounded domain with 퐶∞ boundary which is strongly pseudoconvex, then The Mapping Theorem and Bergman Kernel 퐴(푧, 푤) In the following years Charlie made a number of other (5) 퐾 (푧, 푤) = + 퐵(푧, 푤) log 푄(푧, 푤). Ω 푛+1 notable advances. Any list would have to include his (푄(푧, 푤)) version of Carleson’s theorem about the convergence of Here 푄(푧, 푤) is the holomorphic part of the second- Fourier series, his solution with R. Beals of an outstanding order Taylor expansion of a defining function 휌 of local solvability problem for linear partial differential the domain Ω; 퐴 and 퐵 are smooth functions, with

December 2017 Notices of the AMS 1259 The proof of the lemma requires again a highly involved argument, because, among other things, it is imperative to overcome the obstacle of the log term in (5). When finally all this is done the theorem is: ∞ Let Ω1 and Ω2 be a pair of bounded domains with 퐶 boundaries that are strongly pseudoconvex. Suppose there is a biholomorphism between them. Then this extends to a ∞ 퐶 diffeomorphism between Ω̄ 1 and Ω̄ 2. After his paper (1974) various alternate approaches, simplifications, or extensions were obtained by others, including: L. Boutet de Monvel and J. Sjöstrand (1976); S. Bell and E. Ligocka (1980); L. Nirenberg, S. Webster, and P. Yang (1980); S. Bell and D. Catlin (1982); and F. Forster- nic (1992). Still, more than forty years later, Charlie’s achievement stands as a milestone in the development of several complex variables. Terence Tao

Connecting Fourier Analysis and Geometry During the early 1970s Fefferman made a number of important and fundamental contributions in the theory of oscillatory singular integrals, the study of which can be motivated by the classical problem of determining the nature of convergence of Fourier series and Fourier integrals and which has since had remarkable connections and applications to many other fields of mathematics, including partial differential equations, analytic number theory, and geometric measure theory. To motivate the subject, let us first work in the simplest setting of Fourier series on the unit circle ℝ/ℤ. Given any absolutely integrable function 푓 ∶ ℝ/ℤ → ℂ, one can form Studio Interior, painted by Charlie circa 1968. the Fourier coefficients 푓(푛)̂ for any integer 푛 by the formula 푓(푛)̂ ∶= ∫ 푓(푥)푒−2휋푖푛푥 푑푥, 퐴(푧, 푧) ≠ 0. It should be noted that in the case Ω is the ℝ/ℤ 2 unit ball, we can take 휌(푧) = 1 − |푧| , 푄(푧, 푤) = 1 − 푧 ̄푤, and then the Fourier inversion formula asserts that one and 퐴(푧, 푤) ≡ 푐푛, 퐵(푧, 푤) ≡ 0. Moreover, it should be should have the identity stressed that no such general result has been proved (or ∞ even formulated!) if one drops the assumption of strong (6) 푓(푥) = ∑ 푓(푛)푒̂ 2휋푖푛푥. pseudoconvexity. 푛=−∞ To prove (5) required that at each boundary point If 푓 is sufficiently regular, then there is no difficulty in 푤 ∈ 휕Ω one osculate to a high degree a version of a interpreting and proving this identity; for instance, if 푓 complex ball and transplant its explicit Bergman kernel is continuously twice differentiable, one can show that to Ω. This gave a first-order approximation to 퐾Ω, which the Fourier coefficients 푓̂ are absolutely summable and then had to be followed by a highly intricate iterative that the series appearing on the right-hand side of (6) procedure to ultimately obtain (5). converges uniformly to 푓. If 푓 is instead assumed to Once one has (5) and the Bergman metric is controlled, be square-integrable, then the Fourier coefficients need one can proceed to the main lemma: suppose 푋(푡, 푃0, 휉) not be absolutely summable any more, but it follows is the point of the geodesic starting at 푃0 in the unit easily from Plancherel’s theorem that the series in (6) still direction 휉 and at time 푡. Assume that for some 휉0 the converges unconditionally to 푓 within the Hilbert space nonnegative time portion on the geodesic 푋(푡, 푃표, 휉0) does 퐿2(ℝ/ℤ). not lie in a compact set. Then The situation becomes more subtle if one weakens the (i) lim푡→∞ 푋(푡, 푃0, 휉0) converges to a boundary point regularity hypotheses on 푓 or asks for stronger notions of of Ω. convergence. Suppose for instance that 푓 is an arbitrary th (ii) The same is true for all 휉 near 휉0, and the re- 푝 -power integrable function for some 1 < 푝 < ∞ not sulting mapping of 휉 ↦ 푋(∞, 푃0, 휉) is a local diffeomorphism. Terence Tao is professor of mathematics at the University of (iii) All boundary points can be reached this way. California, Los Angeles. His email address is [email protected].

1260 Notices of the AMS Volume 64, Number 11 equal to 2; thus 푓 ∈ 퐿푝(ℝ/ℤ). As it turns out, the series on the right-hand side of (6) will not, in general, be unconditionally convergent or absolutely convergent in 퐿푝(ℝ/ℤ); however, the partial sums 푁 ̂ 2휋푖푛푥 푆푁(푓)(푥) ∶= ∑ 푓(푛)푒 푛=−푁 will still converge in 퐿푝(ℝ/ℤ) norm to 푓. Establishing this fact is equivalent (by the uniform boundedness princi- ple) to demonstrating that operators 푆푁 are uniformly 푝 bounded in 퐿 (ℝ/ℤ). The operators 푆푁 can be explicitly described as an integral operator: 1 sin((푁 + 2 )푡) 푆푁푓(푥) = ∫ 푓(푥 − 푡) 푑푡. ℝ/ℤ sin(푡/2)

1 sin((푁+ 2 )푡) The kernel sin(푡/2) has a numerator which oscillates rapidly when 푁 is large and a denominator that goes to zero as 푡 goes to zero. As such it is a simple example of an oscillatory singular integral operator (though in this case, it is technically not singular because the numerator also vanishes at 푡 = 0, though its derivative is quite large at that point). There are by now many techniques to establish the required boundedness for 1 < 푝 < ∞; for instance, one can use the Calderón–Zygmund theory of singular integral operators. On the other hand, boundedness (and hence convergence) in 퐿푝 norm is known to fail when 푝 = 1 or 푝 = ∞. A more difficult question is whether the partial sums 푆푁(푓) converge pointwise almost everywhere to 푓. (One can construct examples to show that pointwise every- where convergence can fail, even if 푓 is assumed to be continuous.) This turns out to basically be equivalent to es- Charlie with his wife, Julie, in front of their home in tablishing the boundedness (or more precisely, weak-type Princeton circa 1980. boundedness) in 퐿푝 of not just each individual operator 푆푁, but the more complicated Carleson maximal operator observed that if 풞푓(푥) ∶= sup |푆푁(푓)(푥)|. 푁>0 one replaced the an important Controlling this operator is substantially more difficult ball with a cube, then a modifica- than controlling a single 푆푁, but this was famously connection between achieved in 1966 by Lennart Carleson (for 푝 = 2) and then tion of the one- by Richard Hunt (for general 1 < 푝 < ∞). On the other dimensional the- such Fourier-analytic hand, a famous construction of Kolmogorov produces an ory allowed one 푝 absolutely integrable function 푓 whose partial Fourier se- to establish 퐿 problems and boundedness and ries 푆푁(푓) diverges pointwise almost everywhere (or even questions in everywhere, if one is more careful in the construction). convergence of Fefferman made multiple contributions to these ques- these multipliers geometric measure tions and their higher-dimensional analogues; we shall for all 1 < 푝 < ∞. restrict our attention here to just two of his most well- On the other hand, theory known and influential works in this area. Firstly, for he showed the sur- higher-dimensional Fourier series, if the circle ℝ/ℤ is prising fact that replaced by a torus (ℝ/ℤ)푑 for some 푑 ≥ 2, Fefferman the ball multipliers were not bounded uniformly on 푝 studied the ball multiplier 퐿 for any 푝 ≠ 2, which implied in particular that one could construct functions in 퐿푝((ℝ/ℤ)푑) whose partial ̂ 2휋푖푥⋅푛 푆푁(푓)(푥) ∶= ∑ 푓(푛)푒 푑푥, 푝 Fourier series 푆푁(푓) did not converge in 퐿 norm! We 푑 푛∈ℤ ∶|푛|≤푁 can describe a modern version of Fefferman’s remarkable where |푛| denotes the Euclidean norm of 푛; thus 푆푁 sums construction here in the two-dimensional case 푑 = 2. the terms of the Fourier series whose frequency 푛 lies This construction revealed an important connection be- in the ball of radius 푁. On the one hand, Fefferman tween such Fourier-analytic problems and questions in

December 2017 Notices of the AMS 1261 geometric measure theory and in particular the Kakeya sums such as needle problem, which asked one to find the minimal area 푁 2휋푖(훼 푛+훼 푛2+⋯+훼 푛푘) subset of the plane in which one could rotate a unit line ∑ 푒 1 2 푘 segment (or “needle”) by a full rotation. In 1919 Abram 푛=1 Besicovitch gave a construction that implied that such a as one varies the frequencies 훼1, … , 훼푘 and which has rotation could be carried out in a set of arbitrarily small important consequences in analytic number theory. measure. A modification of this construction produces, In 1973 Fefferman revisited Carleson’s theorem on the pointwise convergence of Fourier series and gave a strik- for any given 휀 > 0, a family of rectangles 푅 , … , 푅 which 1 푛 ing new proof that was a key impetus for the modern field overlapped heavily in the sense that the measure of the of time-frequency analysis. Whereas Carleson’s argument union of the 푅 was less than 휖 times the sum of the 푖 focused on carefully decomposing the function 푓, Feffer- individual measures, but such that if one translated each man’s strategy proceeded by instead decomposing the rectangle 푅푖 along its long axis by (say) three times its operator 풞. The first step was to linearize this operator ̃ ̃ length, the resulting rectangles 푅1,…, 푅푛 were disjoint. by replacing it with the infinite family of operators On the other hand, after suitable rescaling of this family 푆 (푓)(푥) ∶= 푆 푓(푥) of rectangles and taking 푁 large enough, it is possible 푁(⋅) 푁(푥) to construct a wave packet 푓 supported on each shifted sin((푁(푥) + 1 )푡) 푖 = ∫ 푓(푥 − 푡) 2 푑푡, rectangle 푅̃푖 such that the two-dimensional ball multiplier ℝ/ℤ sin(푡/2) 푆푁, when applied to 푓푖, became large on the rectangle 푅푖. where 푁 ranged over all measurable functions 푁 ∶ ℝ/ℤ → By testing 푆푁 on a suitable linear combination of these ℤ from the circle to the integers. As no regularity hy- wave packets 푓푖, one can demonstrate the unboundedness potheses are placed on the function 푁, this integral of 푆푁 for any 푝 > 2, and a duality argument then handles operator is very rough and seemingly hopeless to attack the remaining case 푝 < 2. by conventional harmonic analysis methods. However, The fundamental connection between higher- guided by key examples of these functions 푁, Fefferman dimensional Fourier analysis and Kakeya-type questions realized that the graph of the function 푁 (viewed as a has guided much of the subsequent work in the area; it subset of phase space ℝ/ℤ × ℤ) could be used as a sort is now standard practice to control oscillatory integral of road map to efficiently decompose the operator 푆푁(⋅) into components indexed by phase space rectangles (or operators (such as 푆푁) by first decomposing the functions tiles), which could then be organized into trees and then involved into wave packets such as the functions 푓푖 forests, the contributions of which could be estimated by mentioned above, apply the operators to each wave a combination of clever combinatorial arguments and the packet individually, and use a combination of Fourier almost orthogonality of various pieces of the operator. analytic methods and geometric analysis to control the This highly original argument took some time to be prop- superposition of these operators. A recent triumph of erly integrated with the rest of the subject, but through these sets of techniques has been the resolution last year the work of Michael Lacey, Christoph Thiele, and others, of the Vinogradov main conjecture in analytic number starting in the late 1990s, the techniques of Carleson and theory by Bourgain, Demeter, and Guth, which gives Fefferman were unified with more traditional tools from near-optimal bounds on the mean values of exponential Calderón–Zygmund theory to obtain a systematic set of methods in time frequency analysis to control highly singular operators that had previously been out of reach, Selected Honors and Distinctions of such as the bilinear Hilbert transform of Calderón, as Charles Fefferman well as many variants of relevance to ergodic theory or to Raphael Salem Prize 1971 scattering theory. American Academy of Arts and Sciences 1972 Alan T. Waterman Award 1976 Louis Nirenberg Fields Medal 1978 National Academy of Sciences 1979 Charlie Fefferman on PDEs Doctor Honoris causa: University of Maryland 1979 As is well known, Charlie Fefferman was a child prodigy: American Philosophical Society 1985 when he was in fourth grade he read, and understood, Doctor Honoris causa: Universidad Autónoma de math books going up to calculus. He received his PhD at Madrid 1990 Princeton at age twenty. His adviser was Eli Stein, also a wonderful mathematician and mentor. Bergman Prize 1992 I first met Charlie around 1972, shortly after he was Bôcher Prize 2008a made full professor at the University of Chicago at age Wolf Prize 2017 twenty-two—the youngest full professor anywhere. I still remember listening with great pleasure to his invited a See article in April 2008 Notices, www.ams.org/notices /200804/tx080400499p.pdf. Louis Nirenberg is emeritus professor of mathematics at New York University. His email address is [email protected].

1262 Notices of the AMS Volume 64, Number 11 microlocal analysis and PDEs. In 1985 Charlie wrote a beautiful expository paper on “the uncertainty principle” describing some of their work. It contains a wealth of deep results; I recommend that all students studying PDE read it. If a function 푢 is mainly concentrated in a box 푄 and its Fourier transform is concentrated in a box 푄′, one says that 푢 is microlocalized in 푄♯푄′. The uncertainty principle says, essentially, that |푄| ⋅ |푄′| ≥ 1. They study more complicated regions 퐵푎 (than the cubes) and decompose 퐿2 functions into a sum of components microlocalized using the 퐵푎. The decomposition is used to diagonalize pseudodifferential operators modulo small errors. There are applications to solvability, …why states to fundamental solutions, and to Schrödinger equations. Con- of matter, nections with Egorov’s theorem such as and quantum theory are given. Using this a pseudodifferential molecules, operator is reduced to a multi- plier. Their work also connects form at with symplectic geometry. It is simply striking. suitable tem- Charlie has written many deep papers on mathematical peratures… physics. A long one, in 1986, studies the quantum mechanics of 푁 electrons and 푀 nuclei. He tries to understand why states of matter, such as molecules, form at suitable temperatures. He obtains very striking results, but the general problem is still open. Things depend on the lowest eigenvalue of a Hamilton- ian. With L. Seco he wrote a series of interesting papers. Louis Nirenberg (seated), Charlie, and Elias Stein at a They obtain asymptotic formulas for the ground state celebration of Nirenberg’s Abel Prize, Courant of a nonrelativistic atom. Very refined estimates for the Institute, New York, circa 2015. eigenvalues and eigenfunctions of an ODE are obtained. Charlie also wrote a number of interesting papers in fluid dynamics with various co-authors. With Donnelly address at the International Congress in Vancouver in he wrote a lovely paper on nodal sets for real eigenfunc- 1974 on recent progress in classical Fourier analysis. tions 퐹 satisfying Δ퐹 + 푎퐹 = 0 on a compact connected Charlie has made fundamental contributions in an Riemannian manifold. Near the nodal set 푁 (where 퐹 enormously wide range of subjects: harmonic analysis— vanishes) they prove that 퐹 vanishes on 푁 to at most 1/2 he’s a world master; analysis on complex manifolds; order 푐|푎| . The paper contains a variety of results, microlocal analysis and linear partial differential (and including behavior near zeros of holomorphic functions pseudodifferential) operators; quantum mechanics; fluid in higher dimensions. flow, Euler, and Navier-Stokes equations; and generalizing I would like to add that I consider Charlie one of the Whitney’s extension theorem. deepest and most brilliant mathematicians I have ever I will confine myself to a few topics connected with met. He is also an excellent speaker, and his papers are a partial differential equations and related things. In 1971 pleasure to read. He and his wife, Julie, have been close Charlie proved that the space of functions BMO is the friends all these years. dual to the Hardy space 퐻1. F. Treves and I presented a condition, 푃, which we conjectured would be necessary and sufficient for local solvability of linear PDEs of principal type. We proved this in the case that the real and imaginary parts of the leading coefficients are real analytic. Shortly afterwards Charlie and R. Beals proved the sufficiency of 푃 in the nonanalytic case. Charlie, in collaboration with A. Córdoba and D. H. Phong, wrote a series of deep papers connected with

December 2017 Notices of the AMS 1263 Joseph J. Kohn formal approximate solutions, and with these, using the smoothness of biholomorphic mappings, he actually 2 Fefferman’s Contributions to the Theory of computes the metric 푑푠 and its associated Hamiltonian. Several Complex Variables He uses this to show some surprising behavior of chains. Fefferman’s study of these problems continues in the Charles L. Fefferman has made numerous fundamental monumental paper “Parabolic invariant theory in complex contributions to the theory of several complex variables analysis.” His analysis starts with an analogy between Rie- (SCV). His published papers have had a major impact mannian manifolds 푀 and strictly pseudoconvex domains on the field. His expository writing, his lectures, and his discussions have provided students, colleagues, and 퐷. An isometry of 푀 in normal coordinates is a rotation collaborators with much inspiration and deep insights in 푂(푛). Thus the problem of determining whether two into the subject. He obtained many seminal results in SCV Riemnnian manifolds are isometric is reduced to finite covering a large range of topics; it is not possible to give a dimensions. In Fefferman’s setting the analogue of 푂(푛) is the group 퐻+ of all linear fractional transformations coherent description of all these within the limits of this 푛 article. Here I will briefly describe some of the highlights of ℂ which preserve 휕퐷0, where 퐷0 is an approximation ̃ of his groundbreaking work. of 퐷. If Φ ∶ 퐷 → 퐷 is a biholomorphism, then the Moser normal form at 푝 ∈ 휕퐷 corresponds to the Moser normal Fefferman’s first contribution to complex analysis was + a remarkable achievement: He proved boundary regularity form at Φ(푝) by an element of 퐻 . So, analogously to the of holomorphic mappings using the Bergman metric. His Riemannian case, the problem of deciding whether two proof is highly original, it is a tour-de-force. It is described strictly convex domains are biholomorphic is reduced to in more detail by Stein above. finite dimensions. This leads to certain invariant polyno- One of the principal themes in SCV is the study mials related to the Moser normal form. The Bergman of domains of holomorphy in ℂ푛. Let 퐷 ⊂ ℂ푛 be a kernel on 퐷 is analogous to a heatlike kernel, 퐾푡(푥, 푦), on domain with smooth boundary 휕퐷. Then it is a domain 푀. This 퐾푡 has an asymptotic expansion, and 퐾푡(푥, 푥) = − 푛 푘 of holomorphy if and only if it is pseudoconvex, which 푐푛푡 2 ⋅ {1 + ∑푘+1 훾푘(푥)푡 }, where the 훾푘 are invariants means that the Levi form is nonnegative. Great progress determined by the Riemannian metric. Analogously for 푛+1 ̃ has been made in the study of the analysis and geometry 퐷 = {휓 > 0}, 퐾퐷(푧, 푧) = 휙(푧)/휓 +휙(푧)log휓. The Tay- 푛+1 associated with strictly pseudoconvex domains (that is, lor expansions of 휙̃ and 휙 modulo 푂(휓 ) are uniquely when the Levi form is positive definite), much of it due to determined by the Taylor expansion of 휕퐷. To carry out Fefferman’s groundbreaking research. In “The Bergman the analogy between the heat kernel and the Bergman kernel and biholomorphic mappings of pseudoconvex kernel, Fefferman finds a function which is the analogue domains,” Fefferman proves a fundamental result: if of 푡. This analogue is an approximate formal solution of 퐹 ∶ 퐷1 → 퐷2 is a biholomorphic map between two strictly the Monge-Ampère equation 퐽(푢) = 1. To complete the pseudoconvex domains, then it is smooth up to the analysis Fefferman must overcome enormous difficulties boundary of 퐷1. This result is a generalization of the which arise from the fact that while Weyl’s analysis is Riemann mapping theorem; it enables one to attach based on the semisimple group 푂(푛), Fefferman has to invariants to the boundary points, providing a powerful deal with the group 퐻+, which is not semisimple. This hold on the classification problem. Fefferman’s proof of is not only an impressive technical feat but it introduces this result involves a profound analysis of the Bergman seminal original methods to the subject. kernel and metric; it is sketched in Stein’s section. Among Fefferman’s many important contributions to Fefferman continues his study of strictly pseudocon- the study of strictly pseudoconvex domains is his 1985 vex domains in “Monge-Ampère equations, the Bergman joint paper with Harold Donnelly, “Fixed point formula kernel, and geometry of pseudoconvex domains.” Here for the Bergman kernel.” They prove the following. Let he studies their geometry, in particular the behavior of Ω be a strictly pseudoconvex domain and 퐾(푧, 푤) the chains—curves on the boundary that are preserved by associated Bergman kernel. Suppose that 훾 ∶ Ω → Ω is a biholomorphic maps; they are analogous to geodesics holomorphic automorphism having no fixed points on 휕Ω. in Riemannian geometry. To compute the Bergman met- Then the fixed point set of 훾 consists of a finite number ric and its associated Hamiltonian, Fefferman considers of points 푝1, 푝2, … , 푝푘. Denote by 훾∗푗 the holomorphic the Dirichlet problem for the following Monge-Ampère differential of 훾 at the point 푝푗. Let 퐼 denote the identity equation: endomorphism and 퐽훾(푧) the holomorphic Jacobian of 훾. Then 푛 푢 푢푘̄ 퐽(푢) = (−1) det ( ) = 1 (−1)푛 푢푗 푢 ̄ 푗푘 푖≤푗,푘≤푛 ∫ 퐾(푧, 훾푧)퐽훾(푧)푑푧 = ∑ . Ω 푗 푑푒푡(퐼 − 훾∗푗) in 퐷 with 푢 = 0 on 휕퐷. Then the Bergman kernel −(푛+1) is replaced by 퐶푛(푢(푧)) with 푢 as above. The This formula is then applied to study circle actions on Ω. difficulty here is that it is not known whether such a 푢 Fefferman has made several contributions to geometry exists. Nevertheless, Fefferman finds a method of finding which are relevant to the understanding of CR structures, as in the 2003 joint paper with Kengo Hirachi, “Ambient Joseph J. Kohn is emeritus professor of mathematics at Princeton metric construction of Q-curvature in conformal and University. His email address is [email protected]. CR geometries.” All the work described above concerns

1264 Notices of the AMS Volume 64, Number 11 the study of strictly pseudoconvex domains and CR manifolds. He has also made major contributions to the weakly convex case, which are described briefly below. The local analysis of strictly pseudoconvex domains and CR manifolds is driven by the approximation of 퐷 by 퐷0 mentioned above. In the general pseudoconvex case there is no analogous method. I have been privileged to collaborate with Fefferman in working on some of these problems. In a beautiful expository paper, “Kohn’s microlocalization of 휕̄ problems,” Fefferman gives the background for our results. Our work deals with Hölder estimates for the operators 휕̄ and 휕푏̄ , the associated Laplacians, and projection operators. The starting point for the analysis of 휕푏̄ on 휕퐷 and on CR manifolds is the local subelliptic estimate 2 ̄ 2 ∗̄ 2 ‖휑‖휀 ≤ 퐶(‖휕푏휑‖ + ‖휕푏 휑‖ ), where the 휑 are (0, 1)-forms in 퐶∞(푈) in an open set 푈. For 0 C. Robin Graham, Charlie Fefferman, and Sun-Yung the analysis of 휕̄the starting point is an analogous estimate ∞ Alice Chang in 2009 at a conference in Charlie’s on 퐶0 (푈 ∩ 퐷) with appropriate boundary conditions honor. on 푈 ∩ 휕퐷. These estimates imply 퐶∞ regularity. The problem is to prove Hölder regularity. This regularity is proved in the special case where the Levi form is locally efforts to understand the asymptotic expansion of the diagonalizable, which means that the Levi form can be Bergman kernel. As Charlie had previously shown, the expressed in terms of local tangential (1, 0) vector fields restriction to the diagonal of the Bergman kernel 퐾 of a 퐿 , … , 퐿 by 푐 = ℒ(퐿 , 퐿̄ ), and it is diagonalizable if 1 푛−1 푖푗 푖 푗 smooth, bounded strictly pseudoconvex domain Ω ⊂ ℂ푛 there exist 퐿s so that 푐 = 휆 훿 . Diagonalizability clearly 푖푗 푖 푖푗 can be written in the form holds when 푛 = 2, but in case 푛 > 2 it is a severe restriction. This work and an extension to the three- 퐾(푧, 푧) = 휑(푧)휌(푧)−푛−1 + 휓(푧) log 휌(푧), dimensional case is contained in a collection of joint where 휌 is a smooth defining function for 휕Ω and 휑, papers (one including M. Machedon) published in 1988– 휓 ∈ 퐶∞(Ω). This expansion can be viewed by analogy 1990. In the general nondiagonalizable case it remains to the asymptotic expansion of the heat kernel of a a difficult problem to prove Hölder estimates. The first Riemannian manifold restricted to the diagonal. The obstacle is that in these cases the subelliptic estimates heat kernel can be expanded in powers of the time do not fit the CR structure, and thus different and much variable 푡, and the coefficients in the expansion are stronger estimates are needed for a starting point. In local scalar invariants of Riemannian metrics, which Fefferman’s remarkable paper “The uncertainty principle” can be constructed as contractions of tensor products and in subsequent work with D. H. Phong, powerful of covariant derivatives of the curvature tensor. The new methods of proving a priori estimates in PDE are Bergman kernel on the diagonal is determined locally by established. These give insights into the type of problems the boundary up to a smooth function, so it was natural discussed here and, hopefully, will lead to a solution. to try to find an analogous expansion for 휑 to order 푛 + 1 and for 휓 to infinite order. But several problems Sun-Yung Alice Chang and immediately arose in contemplating carrying this out. C. Robin Graham One was that Ω is not canonically a product near 휕Ω, so there was no obvious analogue of 푡, nor was there Fefferman’s Work in Conformal Geometry an obvious way to formulate an expansion in such a As Charlie’s colleagues and collaborators, we feel priv- way that the coefficients would be geometric invariants ileged to have had the opportunity to work with him. of the boundary. But even if one could surmount these His contributions to conformal geometry are centered difficulties, the most glaring problem was the fact that it around the ambient metric construction. We begin with was not known how to construct general scalar invariants a discussion of some background leading up to its of CR structures, the geometric structures induced on nondegenerate hypersurfaces by the complex structure discovery. 푛 The conformal ambient metric grew out of Charlie’s on the background ℂ . work in several complex variables, in particular in his Charlie resolved these difficulties in his groundbreak- ing paper “Parabolic invariant theory in complex analysis.” Sun-Yung Alice Chang is professor of mathematics at Princeton His solution was to construct a Lorentz signature, asymp- ∗ University. Her email address is [email protected]. totically Kähler-Einstein metric 푔̃ on ℂ × Ω, where ∗ C. Robin Graham is professor of mathematics at the University of ℂ = ℂ\{0}, via a formal solution of a degenerate complex Washington. His email address is [email protected]. Monge-Ampère equation. This metric 푔̃ is invariant under

December 2017 Notices of the AMS 1265 rotations and homogeneous under dilations in ℂ∗ and sheet of the hyperboloid arising as the −1-level set of is invariantly associated to the CR geometry on 휕Ω. The the Lorentz signature quadratic form. This construction formal solution to the Monge-Ampère equation plays the generalizes to the case of a general conformal manifold role of the time variable 푡, and scalar CR invariants can as the “conformal infinity”; the ambient metric can be be constructed using the curvature tensor and the Levi– restricted to a natural hypersurface in 풢̃, and the resulting Civita connection of 푔̃ in a manner roughly analogous to Poincaré metric has asymptotically constant negative Ricci the case of Riemannian geometry. There are more details curvature. of this solution in Joseph Kohn’s section. These constructions have been The conformal ambient metric, introduced in joint enormously influential; the ambi- work with C. R. Graham, “Conformal invariants,” and The ent and Poincaré metrics are now also denoted 푔̃, is an analogue in a different setting: viewed as fundamental in confor- it is determined by the datum of a conformal class ambient mal geometry and beyond. As (푀, [푔]) of Riemannian metrics on a manifold 푀 of metric described above, one of Charlie’s dimension 푛 ≥ 3. Metrics in the conformal class are original motivations for the con- sections of a ray subbundle 풢 of the bundle of symmetric opened up struction in CR geometry was to 2-tensors on 푀, and 푔̃ is a Lorentz signature metric on the construct and characterize scalar ambient space 풢̃ = 풢×ℝ determined asymptotically along new arenas invariants of CR structures to de- 풢 ≅ 풢 × {0}. The model is the sphere 푆푛, whose group scribe the asymptotic expansion of conformal motions is the Lorentz group 푂(푛 + 1, 1) for study in of the Bergman kernel. The am- of linear transformations of ℝ푛+2 preserving a quadratic geometric bient metric enables construction form of signature (푛 + 1, 1). The ray bundle 풢 can be of scalar conformal invariants as identified with the forward pointing half of the null cone analysis. Weyl invariants, constructed as of the quadratic form. The ambient metric for the sphere linear combinations of complete is just the Minkowski metric on ℝ푛+2, which is clearly contractions of covariant deriva- preserved by the conformal motions in 푂(푛+1, 1) in their tives of the curvature tensor of the ambient metric. linear action on ℝ푛+2. For a general conformal class of Determining the extent to which all invariants arise by metrics (푀, [푔]), the ambient metric is a Lorentz signature this construction involved developing a new parabolic metric on 풢̃ determined asymptotically along 풢 × {0} by invariant theory; this was carried out in joint work with the following three conditions: Graham and by Graham with Bailey and Eastwood. The ambient metric opened up new arenas for study (i) 푔̃ is homogeneous of degree two with respect to in geometric analysis. Conformally invariant powers of natural dilations on 풢̃, ∗ the Laplacian were constructed in terms of the ambient (ii) 휄 푔̃ = g0, metric, leading to Branson’s construction of 푄-curvature. (iii) Ric(푔)̃ vanishes asymptotically at 풢 × {0}. 푄-curvature is a higher-dimensional version of scalar Here g0 is a tautological symmetric 2-tensor on 풢 deter- curvature in dimension 2. Branson’s original definition mined by the conformal class, and 휄 ∶ 풢 → 풢×{0} ⊂ 풢×ℝ proceeded by analytic continuation in the dimension, and is the inclusion. In (iii) the asymptotic order of vanishing 푄 curvature was originally regarded as rather mysterious. is infinite if 푛 is odd and is 푛/2−1 if 푛 is even. The ambient Charlie, in separate papers with Graham and Hirachi in metric 푔̃ is uniquely determined up to diffeomorphism the first years of this century, helped illuminate its nature. by these conditions: to infinite order if 푛 is odd and to 푄 curvature enters into the Gauss–Bonnet integrand in order 푛/2 if 푛 is even. higher dimensions (albeit in a more complicated way than The conformal ambient metric was inspired by Charlie’s the scalar curvature in dimension 2). In dimension 4 and construction in several complex variables, but the rela- modulo the part which is pointwise conformally invariant, tionship is closer than mere analogy: the Kähler–Lorentz the Gauss–Bonnet integrand has fully nonlinear structure metric in the several complex variables construction can under conformal change of metric. The analytic study of be viewed as a special case of a conformal ambient metric. such partial differential equations has become a central One takes the conformal class to be the so-called “Feffer- topic in conformal geometry. man metric” (there are too many of these!) on 휕Ω × 푆1, The fundamental idea underlying the ambient/Poincaré which Charlie had constructed earlier. Although one usu- metric is to study geometry in dimension 푛 by passing ally thinks of conformal geometry as simpler than CR to a different but essentially equivalent description in geometry, by this construction the class of CR structures dimension 푛 + 1 or 푛 + 2. The AdS/CFT correspondence induced on nondegenerate boundaries of domains in ℂ푛 in physics, a major development since its introduction by can be viewed as a subclass of the class of conformal Maldacena in 1997, is based on the same idea. In fact, the structures on even-dimensional manifolds. Poincaré metric construction amounts to the geometry There is a second, equivalent formulation of ambient underlying the AdS/CFT duality between conformal field metrics, namely, as Poincaré metrics. The model here is theories on a boundary at infinity and supergravity in hyperbolic space. Recall that 푆푛 can be viewed as the the bulk. This synergy between geometry and physics has boundary at infinity of hyperbolic space, which arises as stimulated both fields and continues to be a source of the restriction of the Minkowski metric on ℝ푛+2 to one exciting developments today.

1266 Notices of the AMS Volume 64, Number 11 Diego Córdoba

Fefferman on Fluid Dynamics The search for singularities in incompressible fluids has become a major challenge in the area of nonlinear and nonlocal partial differential equations. In particular the existence, or absence, of finite-time singularities with finite energy remains an open problem for 3D incompressible Euler equations in the whole domain ℝ3 or in the periodic 핋3 setting. The local existence in time of classical solutions is well known, and in dimension two these classical solutions exist for all time. A result of Beale, Kato, and Majda asserts that if a singularity forms at time 푇, then the vorticity 휔(푥, 푡) grows so rapidly that 푇 (7) ∫ 푠푢푝푥|휔(푥, 푡)|푑푡 = ∞. 0 Charlie with Antonio Córdoba and Antonio’s son, Notice that in dimension two the vorticity 휔 remains Diego Córdoba, both PhD students of Fefferman. bounded, since it is transported by the flow; this is a main difference with the situation in dimension three. Fefferman’s interest in fluid dynamics started at the Splash and Splat Singularities beginning of the 1990s in a collaboration with Constantin In the case of the incompressible Euler equations with and Majda. They showed that if the velocity remains a free boundary, Charlie, in collaboration with Castro, bounded up to the time of singularity formation, then Gancedo, Gómez-Serrano, and me, established the forma- the vorticity direction 휔(푥,푡) cannot remain uniformly |휔(푥,푡)| tion in finite time of splash and splat singularities for the Lipschitz continuous up to that time. A similar result water wave problem. A splash singularity appears when was proven in the presence of viscosity. This result is the free boundary remains smooth but self-intersects at a extremely useful, since vortex lines are transported by point; a splat singularity is when it self-intersects along an the 3D incompressible Euler flow. Recall that a vortex arc. The difference with the fixed domain case is that the line in a fluid is an arc on an integral curve of the pressure is constant at the boundary, and the boundary vorticity 휔(푥, 푡) for fixed 푡. In numerical simulations of moves with the velocity of the fluid. Therefore finding 3D Euler solutions, one routinely sees that vortex tubes the domain of the fluid is part of the problem. The main (tubular neighborhoods arising as a union of vortex lines) idea of the proof is to choose a conformal map 휙 which grow longer and thinner while bending and twisting. In separates the point of collapse, such that its singular particular, if the thickness of a piece of a vortex tube points (where 휙 cannot be inverted) are located outside becomes zero in finite time, then one has a singular the domain of the fluid. This map transforms the splash solution of 3D Euler. I collaborated with Charlie to prove into a closed curve whose chord-arc is well defined. We that given a criterion only at the level of the velocity field can select an initial velocity that immediately separates then a vortex tube cannot reach zero thickness in finite the point of collapse because the equations are reversible time, unless it bends and twists so violently that no part in time and we can solve backwards in time. In order of it forms a “smooth” tube anymore. If additionally we to return to the original domain and obtain solutions to add the assumption that there is a uniform collapse, in the free boundary incompressible Euler equations, it is the sense that the maximum and the minimum thickness necessary to invert the map 휙. In the presence of viscosity are comparable, then we can obtain a lower bound on the the proof has to be modified since the equations are no rate of decay of the thickness of the tube. longer reversible; we use again the conformal map 휙 A key ingredient in the success of proving the formation that separates the self-intersecting points of the splash of singularities is to identify a scenario in which there is curve. But instead of showing local existence backwards a clear mechanism developing fast vorticity growth, and in time in the transformed domain, we prove local exis- accurate numerics plays a crucial role in this search. Below tence forward in time and show that the solutions depend we describe two plausible singular scenarios discovered stably on the initial conditions. We apply a perturbative first by numerical simulations which led later to a rigorous argument to prove a splash, but not a splat, singularity proof: splash singularities for water waves and shift for Navier–Stokes. stability for the dynamics of the interface between two In recent work Fefferman, in collaboration with Ionescu immiscible incompressible fluids in a porous medium. and Lie, showed that the presence of a second fluid prevents the formation in finite time of both splash and Diego Córdoba is professor of mathematics at the Spanish splat singularities. The condition that the densities of the ∞ National Research Council (CSIC-ICMAT). His email address is fluids are positive is used to show a critical 퐿 bound [email protected]. for the measure of the vorticity in the boundary which

December 2017 Notices of the AMS 1267 prevents self-intersections of the boundary between the fluids.

Shift of Stability and Breakdown Charlie has also worked on the dynamics of the interface between two incompressible viscous fluids with different densities in a porous medium, which is modeled by Darcy’s law. This problem is also known as the Muskat problem. If the heavier fluid is below the light fluid the system is well posed in a Sobolev space 퐻푘, but if the interface is not a graph the system is unstable. Together with collaborators Castro, Gancedo, and López-Fernández we have proven that there exists a nonempty open set of initial data in the stable regime (heavier fluid below the light fluid for which the interface is initially in 퐻4), such that the solution of the Muskat problem becomes immediately real-analytic and then passes to the unstable regime in finite time. Bo’az Klartag and Charlie at Princeton circa 2015. Moreover, the Cauchy–Kowalewski theorem shows that a real-analytic Muskat solution continues to exist for 푚,1 a short time after the turnover. The interface becomes We first discuss the case of 퐶 -functions, which are 푚 푛 more and more unstable as the turnover progresses. In 퐶 -smooth functions 퐹 ∶ ℝ → ℝ for which the norm 훼 fact, there exist interfaces of the Muskat problem such ‖퐹‖퐶푚,1 ∶= sup max |휕 퐹(푥)| 푛 |훼|≤푚 that after turnover their smoothness breaks down. The 푥∈ℝ (8) |휕훼퐹(푥) − 휕훼퐹(푦)| proof follows by a rigorous analysis of the full nonlinear + sup max problem and establishing analytic continuation of Muskat 푥≠푦∈ℝ푛 |훼|=푚 |푥 − 푦| solutions to the time-varying strip of analyticity. is finite. Here, the multiindex 훼 = (훼1, … , 훼푛) is a vector of nonnegative integers, 휕훼퐹(푥) = ( 휕 )훼1 ⋯ ( 휕 )훼푛 퐹(푥) 휕푥1 휕푥푛 Bo’az Klartag and |훼| = ∑푖 훼푖. In a seminal work from the early 2000s, Charlie Fefferman proved the following finiteness Fefferman’s Work on the Whitney Extension principle, which had been conjectured earlier by Brudnyi Problem and Shvartsman: Given an arbitrary set 퐸 ⊆ ℝ푛 and a function 푓 ∶ 퐸 → ℝ Theorem 1. Let 푛 ≥ 1, 푚 ≥ 0, and let 퐸 ⊆ ℝ푛 and 푓 ∶ 퐸 → and an 푚 ≥ 1, does there exist a 퐶푚-smooth function ℝ be arbitrary. Assume that there exists 푀 > 0 with the 푛 퐹 ∶ ℝ → ℝ that extends 푓 (i.e., 퐹|퐸 = 푓)? A problem following property: For any finite subset 푆 ⊆ 퐸 of size at attributed to Hassler Whitney seeks to find plausible most 푘(푚, 푛) < ∞, the function 푓|푆 extends to an auxiliary 푚,1 푛 necessary and sufficient conditions for the feasibility of 퐶 -function 퐹푆 ∶ ℝ → ℝ with ‖퐹푆‖퐶푚,1 ≤ 푀. Then there 푚,1 푛 such a 퐶푚-extension. exists a 퐶 -function 퐹 ∶ ℝ → ℝ with 퐹|퐸 = 푓. Moreover, In the 1930s Whitney proved that the function 푓 ‖퐹‖퐶푚,1 ≤ 퐶푀 for some constant 퐶 = 퐶(푚, 푛). 푚 extends to a 퐶 -smooth function on the entire real line if Thus in order to tell whether 푓 extends to a 퐶푚,1- and only if for any accumulation point 푥 of the set 퐸, the function on the entire ℝ푛, it suffices to consider subsets difference quotient of order 푚, of 퐸 with at most 푘(푚, 푛) elements. The optimal finiteness constant 푘(푚, 푛) is not known in general. 푚 1 [푥0, … , 푥푚](푓) ∶= 푚! ⋅ ∑ ⎛∏ ⎞ 푓(푥푖), Merely a year or two following his proof of Theorem 푥 − 푥 푖=0 ⎝푗≠푖 푖 푗 ⎠ 1, Fefferman was able to resolve Whitney’s 퐶푚-extension problem in another remarkable work. The reader is converges to a finite limit as the distinct points 푥0, … , 푥푚 ∈ 퐸 tend to the point 푥. This finite limit will be the 푚th- referred to Fefferman’s 2009 Bulletin article on the topic for a precise formulation of his 퐶푚-theorem, which derivative at 푥 of any 퐶푚-smooth function that extends involves the notion of Glaeser refinement of certain fiber 푓. In fact, [푥 , … , 푥 ](푓) equals the 푚th-derivative of the 0 푚 bundles over an arbitrary set 퐸 ⊆ ℝ푛, where the fiber of Lagrange interpolation polynomial 푃 of degree 푚 with the bundle at a base point 푥 ∈ 퐸 consists of potential 푃(푥 ) = 푓(푥 ) for all 푖. 푖 푖 Taylor polynomials at 푥 of a 퐶푚-extension function. This The extension problem in ℝ푛 for 푛 ≥ 2 is more difficult, form of Fefferman’s solution is closely related to an earlier as there could be many potential candidates for the th work by Bierstone, Milman, and Pawłucki dealing with a 푚 -order Taylor polynomial of the extension function. subanalytic set 퐸. Glaeser himself settled the case 푚 = 1 in the 1950s. Bo’az Klartag is professor of mathematics at Tel Aviv University. The next question is whether one can turn Fefferman’s His email address is [email protected]. constructive proof into an actual algorithm for a nearly

1268 Notices of the AMS Volume 64, Number 11 optimal 퐶푚-interpolation of data. From the viewpoint of 퐶2(ℝ2), the preprocessing time is at most 퐶(휀)푁 log 푁, computer science, the problem may be formulated as the query work is 퐶 log(푁/휀), and the storage 퐶(휀)푁. follows. We are given a large, finite subset 퐸 ⊂ ℝ푛 and a Moreover, Fefferman has also come up with a (1 + 휀)- function 푓 ∶ 퐸 → ℝ. For 푚 ≥ 1 define version of Whitney’s extension theorem, whose proof uses the notion of a “gentle partition of unity.” For ‖푓‖퐶푚(퐸) = inf{‖퐹‖퐶푚(ℝ푛); simplicity, we restrict our attention to the central case of 푛 푚 퐹 ∶ ℝ → ℝ is 퐶 -smooth with 퐹|퐸 = 푓}, a finite set 퐸 in the following formulation of his theorem: 푚 where ‖퐹‖퐶푚(ℝ푛) = ‖퐹‖퐶푚−1,1 for a 퐶 -smooth function Theorem 2. Let 푚, 푛 ≥ 1 and 휀 > 0 be given, and let 퐸 ⊆ 푛 푛 퐹 ∶ ℝ → ℝ. Our goal is to compute, using an (idealized) ℝ be a finite set. Let (푃푥)푥∈퐸 be a family of polynomials of 푛 digital computer, a function 퐹 ∶ ℝ → ℝ with 퐹|퐸 = 푓 degree 푚 in 푛 real variables. # and ‖퐹‖퐶푚(ℝ푛) ≤ 퐶‖푓‖퐶푚(퐸), for 퐶 > 0 being a constant Assume that for any 푆 ⊆ 퐸 with #(푆) ≤ 푘 (푚, 푛, 휀) 푚 푛 depending only on 푚 and 푛. In a joint work with Klartag, there exists an auxiliary 퐶 -smooth function 퐹푆 ∶ ℝ → ℝ th Fefferman has transformed his proof of Theorem 1 into with ‖퐹푆‖퐶푚(ℝ푛) ≤ 1, such that for any 푥 ∈ 푆, the 푚 -order an algorithm with the following interface: Taylor polynomial of 퐹푆 at the point 푥, denoted by 퐽푥(퐹푆), 푚 (i) First, the user enters the entire data set (the coordi- satisfies 퐽푥(퐹푆) = 푃푥. Then there exists a 퐶 -smooth func- 푛 nates of the points of 퐸 and the values of 푓) into the tion 퐹 ∶ ℝ → ℝ with 퐽푥(퐹) = 푃푥 for all 푥 ∈ 퐸 such that computer. The computer then works for a while, per- ‖퐹‖퐶푚(ℝ푛) ≤ 1 + 휀. forming at most 퐶푁 log 푁 operations where 푁 = #(퐸), We could go on and on. Only space limitations prevent after which it signals that it is ready to accept further us from describing additional Whitney-related theorems input. that were published by Fefferman in recent years. We (ii) Then, whenever the user enters the coordinates of a would like to conclude by wishing Charlie and the math- point 푥 ∈ ℝ푛, the computer rapidly responds, using ematical community many more beautiful results on the only 퐶 log 푁 operations with the value 퐹(푥). subject in years to come! The function 퐹 that is computed by the algorithm is Postscript: In June 2017 Fefferman and Shvartsman an- guaranteed to be an extension of 푓, with ‖퐹‖퐶푚(ℝ푛) ≤ nounced their proof of a finiteness principle for Lipschitz 퐶‖푓‖퐶푚(퐸). The storage required by the algorithm is selection in an arbitrary metric space. That is, assume bounded by 퐶푁. Of course, the handling of real numbers that we are given a metric space 푋 and a compact, convex in the digital computer is only up to finite precision; the 푑 set 퐾푥 ⊆ ℝ associated with any point 푥 ∈ 푋. Assume algorithm returns its answers in the given precision of the that for any subset 푆 ⊆ 푋 with #(푆) ≤ 2푑 there exists a digital computer. The reader might wonder why 퐶푁 log 푁 1-Lipschitz map 퐹 ∶ 푆 → ℝ푑 with 퐹 (푥) ∈ 퐾(푥) for all operations suffice when Theorem 1 requires an inspection 푆 푆 푥 ∈ 푆. Then there exists a 퐶-Lipschitz map 퐹 ∶ 푋 → ℝ푑 of all subsets of 퐸 of size 푘(푚, 푛). As it turns out, there with 퐹(푥) ∈ 퐾(푥) for all 푥 ∈ 푋, where 퐶 is a constant exists a list of no more than 퐶푁 subsets, each of size depending only on 푑. no greater than 퐶, that matter, as proven by Fefferman. Extensions of his result exist for smooth selection and for interpolating multidimensional functions with noisy data. Jürg Fröhlich, Luis Seco, and Particularly notable is the extension to Sobolev spaces; Michael Weinstein in a series of papers Fefferman, Israel, and Luli proved the existence of an extension and described an algorithm Charlie’s Romance with Quantum Theory to construct it. That algorithm shares features with the 푚 Our memories of Charlie Fefferman go back to the 퐶 -algorithm described above: the preprocessing time is middle of the seventies when Jürg and Charlie arrived at 퐶푁 log 푁, and the query time is only 퐶 log 푁. Princeton. Barry Simon, then at Princeton, had announced We move on to discuss some of Fefferman’s work on the that he had a “secret weapon” that might well enable (1+휀)-version of the quantitative Whitney problem, where him to construct models of local relativistic quantum one is given a small 휀 > 0 and is seeking a 퐶푚-function 푛 fields in four-dimensional Minkowski space—a dream of 퐹 ∶ ℝ → ℝ extending the given function 푓 ∶ 퐸 → ℝ such many mathematical physicists working in constructive that quantum field theory, yet to come true. It turned out that (9) ‖퐹‖퐶푚(ℝ푛) ≤ (1 + 휀) ⋅ ‖푓‖퐶푚(퐸). Barry’s secret weapon was Charlie! He was convinced The 퐶푚(ℝ푛)-norm is usually considered only up to a that if Charlie were willing to put his mind to problems multiplicative constant; some authors define the 퐶푚 and in relativistic quantum field theory success in solving 퐶푚,1-norms a bit differently, replacing the sum in (8) by a Jürg Fröhlich is professor in the Institute for Theoretical Physics at maximum. Nevertheless, from the point of view of data 푚 The Swiss Federal Institute of Technology (ETH) Zürich. His email interpolation, it makes sense to fix a reasonable 퐶 -norm address is [email protected]. and ask for an efficient algorithm that produces a function Luis Seco is professor of mathematics at the University of Toronto. 푛 퐹 satisfying (9), given a finite set 퐸 ⊆ ℝ and function His email address is [email protected]. values 푓 ∶ 퐸 → ℝ. Michael Weinstein is professor of mathematics and professor of Fefferman devised a polynomial-time algorithm for applied mathematics at Columbia University. His email address carrying out this task for all 푚 and 푛. In the case of is [email protected].

December 2017 Notices of the AMS 1269 them would soon be within reach. Alas, Barry’s hopes Here is what stability of matter means: Consider did not materialize. Luckily we all recovered from this a system in physical space ℝ3 consisting of 푀 static mishap and went on to solve other interesting, albeit nuclei at positions 푦푗 with atomic numbers (charges) 푍푗, somewhat less challenging, problems in quantum theory. 푗 = 1, … , 푀, and 푁 electrons at positions 푥푘, 푘 = 1, … , 푁, For example, Charlie and his former student Antonio interacting through Coulomb forces. The Hamiltonian Córdoba developed the so-called wave-packet transform, operator of such a system is given by an analytical tool built upon Heisenberg’s uncertainty 푁 relations. (10) 퐻푍,푀,푁 = ∑ (−Δ푥푘 ) + 푉푍,푀,푁. There followed a long period when Jürg’s and Charlie’s 푘=1 trajectories did not intersect. In the mid-eighties, after Jürg had moved back to ETH in Zürich, he heard of The Laplacian terms represent the kinetic energy of the an exciting discovery: Charlie had solved the problem electrons (their mass being set to 1/2), while the second of understanding why, at appropriate temperatures and term on the right side is the Coulomb potential energy, densities, atomic nuclei and electrons form gases of which is given by the multiplication operator

neutral atoms or molecules. His 1 푍푗 푍푘 푍푘 result appeared under the ti- (11) 푉푍,푀,푁 = ∑ + ∑ − ∑ . |푥푗 − 푥푘| |푦푗 − 푦푘| |푥푗 − 푦푘| …his tle “The atomic and molecular 푗<푘 푗<푘 푗,푘 nature of matter” in the first distinctive The Hamiltonian 퐻푍,푀,푁 acts on the Hilbert space ℋ푁 issue of the first volume of given by the 푁-fold antisymmetric tensor product of style in a new journal that A. Cór- the one-electron Hilbert space, 퐿2(ℝ3, d3푥) ⊗ ℂ2, where doba had founded, Rev. Mat. 퐿2(ℝ3, d3푥) is the space of orbital wave functions of an analysis: at Iberoamericana. The abstract electron and ℂ2 is the space of its spin states. of the paper reads: “The pur- One also considers the pseudo-relativistic version of pose of this article is to show once the Hamiltonian, with 퐻푍,푀,푁 now given by that electrons and protons, in- powerful, teracting by Coulomb forces 푁 (12) 퐻 = (−Δ )1/2 + 훼 ⋅ 푉 . and governed by quantum sta- 푍,푀,푁 ∑ 푥푘 푍,푀,푁 bare-hands, tistical mechanics at suitable 푘=1 temperature and density, form In this operator the fine structure constant 훼 ≃ 1/137 and elegant a gas of Hydrogen atoms or appears as a fundamental parameter, and (12) comes with molecules.” other mathematical licenses that turn the original physical At around that time, Luis arrived at Princeton as a problem into this form, keeping the essential feature that, graduate student. He was closely familiar with Charlie’s in relativity theory, kinetic energy is proportional to previous work on harmonic analysis, but he had no idea momentum in the high-energy limit. what an electron was. One of the unwritten rules of the Stability of matter is the property that Princeton mathematics department is that it does not offer graduate courses on subjects where there is already (13) ⟨퐻푍,푀,푁휓, 휓⟩ ≥ −퐶푍 ⋅ (푀 + 푁) a good book. As a consequence, professors teach courses in the nonrelativistic case (10), and on whatever topic they just spent their time working on, and it is not unusual that some of their colleagues attend (14) ⟨퐻푍,푀,푁휓, 휓⟩ ≥ 0 the lectures. for the relativistic Hamiltonian (12), where 휓 is an In September 1984 Michael introduced himself to arbitrary wave function in ℋ of norm 1. In the latter Charlie as they were riding on a Fine Hall elevator and 푁 case, the critical role of the fine structure constant said he was new at Princeton. Charlie introduced himself becomes clear after one realizes that in (12), the Coulomb to Michael and said he had been there forever. He gave potential energy and the relativistic kinetic energy scale Michael his coordinates and encouraged him to stop by in the same way, namely, as inverse lengths. Thus, either to chat. That semester Michael followed Charlie’s course, (14) holds or inf⟨퐻 휓, 휓⟩ = −∞. If the bounds (13), advertised in the catalogue as Fourier Analysis on Groups, 푍,푀,푁 which turned out to be about Charlie’s ongoing work on (14), respectively, did not hold true one would conclude stability of matter, treated quantum-mechanically, first that matter must collapse and, in the process, release a established by Dyson and Lenard and later by Lieb and devastating amount of energy. Thirring and others. His course was also attended by Charlie’s first landmark achievement in quantum the- another assistant professor, Rafael de la Llave, who was ory was his work with Rafael de la Llave on stability of Charlie’s collaborator in his work “Relativistic stability matter for Coulomb systems described by (10) and their of matter.” It was through these wonderful lectures pseudorelativistic cousins, as described by (12). Needless that Charlie introduced Michael to deep mathematical to say, Luis understood nothing of all this, except for problems in quantum theory and to his distinctive style one thing: in the relativistic case, Fefferman’s philosophy in analysis: at once powerful, bare-hands, and elegant. was that, since the scaling properties of the operator in And then there was Luis, taking his first course with (12) make the problem hard, one had better exploit these Charlie after arriving at Princeton. properties to one’s advantage by rewriting the kinetic

1270 Notices of the AMS Volume 64, Number 11 energy and the Coulomb potential in a convenient form, multiplied by an oscillatory term depending on 푍. This as follows: paints a number-theoretical perspective on the periodic (15) table of elements. 1 1 1 if both 푥, 0 ∈ 퐵(푧, 푅), 푑푧 푑푅 As these papers on atomic physics were being finished, = ∫ ∫ { } , 5 |푥| 휋 푅>0,푧∈푅3 0 otherwise 푅 in the first half of the nineties, Charlie announced that he would visit ETH. He told Jürg that he would like to meet with a similar expression for the Laplacian term. This him and find out what he was up to. Jürg told Charlie expression is, in some sense, obvious, since the left and that he had a PhD student1 whom he had encouraged to right sides scale in the same way (as an inverse length) and work on a problem that had interested him for well over both are translation invariant. The revolutionary insight a decade: here is that the left-hand side is hard, but the right-hand Consider the quantum-mechanical description of a side is easy: all it requires is counting the number of physical system consisting of 푁 nonrelativistic electrons electrons or nuclei in a ball. Equation (15) turned out with spin- 1 moving in an arbitrary external magnetic field to be a game-changing element, with the strong flavor 2 ⃗ ⃗ ⃗ of microlocal analysis. This attracted Luis to make a 퐵 = curl퐴, where 퐴 is the electromagnetic vector potential ⃗ ⃗ move that would seem absurd to an outsider: to do a (with ∇ ⋅ 퐴 = 0), and in the electro-static field of 푀 nuclei thesis on mathematical physics. Luis’s move began a with atomic numbers 푍푗 ≤ 푍 < ∞, 푗 = 1, 2, … , 푀. Let fruitful collaboration that would last a decade, involved 퐻푍,푀,푁(퐴)⃗ denote the Hamiltonian of this system, which very smart discussions, on-going oversight by de la Llave, is a self-adjoint operator acting on the Hilbert space ℋ푁. and produced a long list of results on Coulomb systems. The magnetic moment of the electron is proportional to Their work concerned single atoms—to be precise, atoms its spin operator. The so-called gyro-magnetic factor, 푔, with large atomic number 푍, setting 푀 = 1 in (10). The of the electron is the factor of proportionality between its ground-state energy of such atoms is given by spin and its magnetic moment; it has the value 푔 = 2, as correctly predicted by Dirac’s equation for a relativistic 퐸(푍, 푁) = min ⟨퐻푍,푁휓, 휓⟩ . 휓∈ℋ푁,‖휓‖=1 electron. It had already been shown by Barry Simon et al. They adopted the asymptotic perspective that 푍 → ∞. that if 푔 were smaller than 2 the system would be stable, While, in nature, 푍 does not get to be much bigger than in the sense that the corresponding Hamiltonian satisfies 100, the Hamiltonian acts on functions that can easily the lower bound 2 depend on several hundred variables. Much of the mathe- (17) 퐻푍,푀,푁(퐴)⃗ > −퐶푍 푁, matical complexity therefore comes from dimensionality ⃗ considerations that will recreate the flavor of the semi- uniformly in the vector potential 퐴, for a 푔-dependent classical limit, WKB theory, spectral asymptotics, and the finite constant 퐶 > 0. Furthermore, if 푔 > 2 the system lattice-point problem of number theory. is unstable, even for a single electron, which is easy to Their first achievement was a proof of asymptotic verify. Nature wants us to understand whether the system neutrality of large ions, proving that the number 푁(푍) of is stable for 푔 = 2! electrons that renders 퐸(푍, 푁) minimal, E. H. Lieb, M. Loss, and H.-T. Yau, partly in collaboration with Jürg, proved in 1985 that for the physical value of 퐸(푍) = 퐸(푍, 푁(푍)) = min 퐸(푍, 푁), 푁 the fine-structure constant 훼, there is a critical value, 푍 , of the atomic number of nuclei with the property satisfies the asymptotic expression crit that, for 푍 > 푍crit, the system is unstable even for only a 0.84 푁(푍) = 푍 + 푂(푍 ). single electron, quite a subtle result. However, at the time The key ingredient was to perform a deep, rigorous of Charlie’s visit in Zurich, it was unknown whether, for benchmarking of the atomic Hamiltonian with its semi- the physical values of the 푔-factor and the fine-structure 1 classical version, the well-known Thomas–Fermi theory. constant, i.e., for 푔 = 2 and 훼 ≃ 137 , stability of matter in This strategy was repeated in several different papers—in magnetic fields holds, i.e., whether there exists a strictly one notable example to obtain asymptotic estimates for positive 푍crit such that, for 푍푗 ≤ 푍 < 푍crit, 푗 = 1, … , 푀, the total spin of an atom—but its climax was reached in the system is stable, in the sense that 1991 when Fefferman and Seco announced their proof (18) 퐻푍,푀,푁(퐴)⃗ > −퐶푍(푁 + 푀), of the Dirac–Schwinger conjecture, a long-standing open problem on the altar of mathematical physics: for a finite constant 퐶푍, uniformly in 퐴⃗, for an arbitrary number of electrons and nuclei. To make some relevant (16) 퐸(푍) = −푐 푍7/3 + 푐 푍2 + 푐 푍5/3 + 표(푍5/3). 푇퐹 푆 퐷푆 steps in the direction of establishing (18) was the project Each term above is loaded with history, and the last Jürg had proposed to his PhD student. The problem was term also signified its last chapter. Its derivation is well known to some of Jürg’s friends at Princeton. When deep and long, involving interesting elements of number Charlie arrived in Zurich, Jürg had not advanced far in un- theory and computer assistance. That number theory raveling the mathematical subtleties hiding beneath (18) becomes relevant in the semiclassical limits is easy to understand: the lattice-point problem is encountered in 1Maxi Seifert—nowadays an expert in the field of mathemati- the spectral asymptotics of the Laplacian. Without this cal finance, the area that Luis also moved into after quantum fact, the term proportional to 푍5/3 in (16) would be mechanics.

December 2017 Notices of the AMS 1271 yet. But he knew that some of Charlie’s techniques might be helpful and he explained the rather poor state of their understanding. Somewhat to his surprise, Charlie had never heard nor thought of this problem, but found the question worth thinking about. Two weeks after Charlie had returned to the US, he sent Jürg copies of hand-written notes containing a proof of (18) for 푍 = 1, assuming that 훼 is small enough. Jürg was deeply impressed by the ideas, which he found very beautiful, and even more by the enormous speed of Charlie’s work. Needless to say, for Jürg and his student, this development was a mixed blessing. Shortly after Charlie had found his proof of “Stability of matter in magnetic fields,” Lieb, Loss, and Solovej discovered a considerably shorter and more elegant proof of this result, using one of Charlie’s ideas. They proved the following variant of (18): 1/3 2/3 (19) 퐻푍,푀,푁 > 퐶푍푁 푀 , Figure 1. Dispersion surfaces of Wallace’s celebrated provided 훼 < 0.06 and 푍 ⋅ 훼2 < 0.04. Thus, the problem 2-band tight-binding model with its conical singular- was settled! ities. Jürg also got more than his share of pleasure in studying systems of electrons and nuclei interacting with the electromagnetic field. For example, in a collaboration with the 퐿2-spectrum is the union over quasi-momenta k in the Charlie, Gian Michele Graf, and a student, they showed Brillouin zone of the discrete (Floquet–Bloch) spectra of that systems of electrons and nuclei interacting with the Hamiltonians 퐻휆(k) = −(∇ + 푖k)2 + 휆2푉(x) on 퐿2(ℝ2/Λ): quantized electromagnetic field are stable, provided an 퐸휆(k) ≤ 퐸휆(k) ≤ ⋯ ≤ 퐸휆(k) ≤ ⋯ . ultraviolet cutoff is imposed on the electromagnetic field. 1 2 푏 휆 Charlie spent the academic year 2007–08 at Columbia The maps k ↦ 퐸푏 (k) are Lipschitz continuous, and their University, during which he gave the Eilenberg Lectures graphs are called dispersion surfaces. Dirac points are 휆 on his seminal work on Whitney problems (described energy-quasimomentum pairs (퐸퐷, K⋆) in a neighborhood 휆 in Klartag’s contribution). During this period, Charlie’s of which two successive dispersion surfaces, 퐸−(k) and 휆 and Michael’s families befriended each other, and the 퐸+(k), have the following behavior for k near K⋆: conversations between Charlie and Michael ranged widely. 휆 휆 휆 퐸±(k) − 퐸 = ± 푣 |k − K⋆| ( 1 + 풪(|k − K⋆|) ) , Eventually they focused on mathematical aspects of an 퐷 퐹 휆 important problem in condensed matter physics: the where 푣퐹 > 0. The dispersion surfaces typically displayed properties of graphene and its artificial analogues. by physicists are not of a Schrödinger operator on the Around that time Michael heard the physicists Horst continuum but rather the 2-band tight-binding (discrete) Stoermer and Philip Kim lecture on the remarkable physi- model of P. R. Wallace from his 1947 pioneering study of cal properties of graphene. Graphene is a nearly perfect graphite; see Figure 1. This model is the workhorse for two-dimensional material, a single atomic layer of carbon condensed matter modeling of graphene. atoms arranged in a honeycomb structure. Just about ev- As their conversations meandered between math and ery lecture on graphene by experimentalists and theorists non-math, Fefferman and Weinstein noted that all hon- begins with figures of its iconic Dirac cones, conical singu- eycomb Schrödinger operators with small (low contrast) larities in the band structure; see, for example, Figure 1. potentials have Dirac points and Wallace’s tight-binding This structure is beautiful for its simplicity (generic band model, likewise, has Dirac points. They wondered whether structures of crystals can be very messy) and profound Dirac points exist for arbitrary, or at least generic, poten- in its physical implications. tials of this class. In a joint 2012 paper in the Journal To understand Dirac cones one considers the of the American Mathematical Society they proved that Schrödinger operator 퐻휆 = −Δ + 휆2푉(x), where 푉 is a they do. Furthermore, for 푉(x) equal to a sum of atomic honeycomb lattice potential. That is, 푉 is real-valued potential wells centered on the honeycomb structure we 휆 휆 휆 and periodic with respect to the equilateral triangular have that ( 퐸±(k) − 퐸퐷 ) /푣퐹 converges onto the two dis- lattice Λ. Furthermore, with respect to some origin persion surfaces of Wallace’s tight-binding model (Figure of coordinates in ℝ2, 푉 is inversion symmetric (even) 1) as 휆 tends to ∞ uniformly in k varying in the Brillouin ∘ 휆 −푐휆 and 120 rotationally invariant. An important example, zone, where 푣퐹 ≈ 푒 > 0 depends on the atomic well. corresponding to the single-electron model of graphene, They further explored the role that Dirac points play in is obtained by taking 푉 to be a sum of translates of a the origin of robust (topologically protected) edge states, fixed atomic potential well, 푉0, centered at the sites of a which are localized transverse to and plane-wave-like regular honeycomb structure. For any periodic potential, parallel to line defects through the honeycomb structure.

1272 Notices of the AMS Volume 64, Number 11 A m e r i c a n M at h e m at i c a l S o c i e t y

Their remarkable stability and propagation properties are the hallmarks of topological insulators. Can one construct such states analytically? These questions led to a sequence of recent papers with J. P. Lee-Thorp; there remains a great deal to explore and understand in this very active area motivated by questions in condensed matter physics, photonics, and other fields. FEATURED TITLES FROM These results are not the end of the story but are a promising beginning. During the past twenty-five years, many interesting results and novel analytical methods have been discovered. But some of the most challenging problems of quantum theory, e.g., the construction of a physically interesting four-dimensional, local, relativistic quantum field theory, in which Charlie’s participation would undoubtedly have made a difference, remain open. Hilbert’s Seventh We can always hope that Charlie will expand his engage- Problem Solutions and Extensions ment with the mathematics of quantum theory and fulfill Robert Tubbs, University of Barry Simon’s hope to become his secret weapon. Colorado, Boulder This exposition is primarily a survey Fefferman’s PhD Students: of the elementary yet subtle inno- Antonio Córdoba The University of Chicago 1974 vations of several mathematicians between 1929 and 1934 that led Elena Prestini The University of Maryland 1976 to partial and then complete solu- Bernard Marshall Princeton University 1977 tions to Hilbert’s Seventh Problem Roberto Moriyon Princeton University 1979 (from the International Congress of Adrian Nachman Princeton University 1980 Mathematicians in Paris, 1900). Antonio Sánchez Calle Princeton University 1983 2016; 94 pages; Softcover; ISBN: 978-93-80250-82-3; List US$28; AMS members US$22.40; Order code HIN/72 Barnwell Hughes Princeton University 1986 Matei Machedon Princeton University 1986 Joseph Gregg Princeton University 1987 Luis Seco Princeton University 1989 Theory of Semigroups Alberto Parmeggiani Princeton University 1993 and Applications Jarosław Wróblewski Princeton University 1993 Kalyan B. Sinha, Jawaharlal Nehru Alejandro Andreotti Princeton University 1994 Centre for Advanced Scientific Diego Córdoba Princeton University 1998 Research, Bangladore, India, and Ronald Howard Princeton University 1999 Sachi Srivastava, University of Delhi South Campus, New Delhi, India Alan Ho Princeton University 2001 Combining the spirit of a textbook Jorge Silva Princeton University 2001 with that of a monograph on the Rami Shakarchi Princeton University 2002 topic of semigroups and their appli- Jose Rodrigo Diez Princeton University 2004 cations, this book will appeal to readers interested in operator the- Spyros Alexakis Princeton University 2005 ory, partial differential equations, harmonic analysis, probability Garving Luli Princeton University 2010 and statistics, and classical and quantum mechanics. Arie Israel Princeton University 2011 2017; 180 pages; Hardcover; ISBN: 978-93-86279-63-7; List US$38; AMS members US$30.40; Order code HIN/73

Image Credits Photo of C. Robin Graham, Charlie Fefferman, and Sun- Yung Alice Chang courtesy of Sun-Yung Alice Chang. VIEW MORE TITLES AT All remaining photos and Figure 1 courtesy of Julie BOOKSTORE.AMS.ORG/HIN

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December 2017 Notices of the AMS 1273