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Home , Avi Wigderson, Charles Fefferman, Claude LeBrun, Dennis Sullivan, Gigliola Staffilani, Harald Bohr

ISSN 0002-9920 (print) ISSN 1088-9477 (online)

of the American Mathematical Society December 2017 Volume 64, Number 11

Ad Honorem page 1254 An Old Problem Resurfaces Nonlocally page 1275 Marcel Berger Remembered page 1285

Notices of the American Mathematical Society December 2017 FEATURED

1254684 1275 26 1285 Ad Honorem Charles An Old Problem Resurfaces Marcel Berger Remembered Fefferman Nonlocally: Gamow's Liquid by Claude LeBrun, Gérard Besson, Misha Gromov, Jim Simons, , Jean- by Antonio CÓrdoba, Elias Stein, Drops Inspire Today's Pierre Bourguignon, , , , Joseph Research and Applications Jacques Lafontaine, Jerry Kazdan, Marie- J. Kohn, Sun-Yung Alice Chang, C. Louise Michelsohn, Pierre Pansu, Robin Graham, Diego CÓrdoba, Bo'az by Rustum Choksi, Cyrill B. Muratov, David Ebin, and Karsten Grove Klartag, JÜrg FrÖhlich, Luis Seco, and and Ihsan Topaloglu Michael Weinstein

We close out 2017 with a celebration of Wolf Prize recipient Charles Fefferman and memories of geometer Marcel Berger. Gamow's liquid model for the atomic nucleus returns in recent research and applications. If hiring, you don't want to miss “Reducing Bias in Faculty Searches.” David Peifer provides the story of Dorothea Rockburne's encounter with at the legendary . —Frank Morgan, Editor-in-Chief

GRADUATE STUDENT SECTION COMMENTARY, CONT'D.

1297 Interview 1322 Book Review: e Real and the Complex Alexander Diaz-Lopez Judith V. Grabiner

1300 WHAT IS... a Scheme? Tom Gannon ALSO IN THIS ISSUE

1302 AMS Graduate Student Blog 1304 Reducing Bias in Faculty Searches Elizabeth A. Burroughs

1313 Dorothea Rockburne and Max Dehn at Black COMMENTARY Mountain College David Peifer 1308 Opinion: Four Goals for Instructors Using Inquiry-Based Learning 1319 Autobiography of Wentsun Wu (1919–2017) Chris Rasmussen, Karen Marrongelle, Oh Nam Kwon, and Angie Hodge 1366 2017 Notices Index

1329 Letters to the Editor 1375 San Diego Meeting Registration Forms Notices A m e r i c A n m AthemAticAl S o c i e t y of the American Mathematical Society A Primer of Mathematical FROM THE AMS SECRETARY Writing Being a Disquisition NO NOT USE 1274 Call for Nominations: D. R. Fulkerson Prize on Having Your Ideas Recorded, Typeset, AVAILABLE IN EBOOK FORMAT Published, Read, and Appreciated, AVAILABLE IN EBOOK FORMAT IN EVERY ISSUE Second Edition AVAILABLE IN EBOOK FORMAT Steven G. Krantz, Washington University, 1327 BookShelf St. Louis, MO Praise for the First Edition:AVAILABLE IN 1330 The Back Page EBOOK FORMAT Krantz, a prolific and distinguished mathe- 1331 Opportunities matical author, discourses engagingly (yet seriously) on the art and etiquette of 1334 Mathematics People virtually all types of writing an academic is likely to encounter… 1336 Inside the AMS Grammatical points, stylistic and typeset- ting issues, and the correct and effective 1338 For Your Information use of mathematical notation are handled deftly and with good humor… [Hopefully] 1341 New Publications Offered by the AMS senior faculty will consider it mandatory reading for graduate students and even 1348 Mathematics Calendar upper-division undergraduates. An enjoy- able way to learn some fundamentals of 1349 Classified Advertising good mathematical writing. Highly recom- mended. 1352 Meetings and Conferences of the AMS —CHOICE This second edition on the modern aspects of mathematical writing features a new, extensive discussion of technological developments. Similar to the first edi- tion, Krantz’s frank and straightforward approach makes this book particularly suitable as a textbook for an undergrad- uate course. 2017; 336 pages; Softcover; ISBN: 978-1-4704-3658-2; List US$45; AMS members US$36; Order code MBK/112

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www.ams.org/wwtbam 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 . 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 of all time, having so far contributed with fundamental results to , 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 under the supervision of Eli Stein. tied, but painting was never in the lead. Eventually he In 1971, at the , 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 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 , 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 , 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 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 ) 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 , 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 . 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 휕푗 ̄푧 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 (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 1971 scattering theory. American Academy of Arts and Sciences 1972 Alan T. Waterman Award 1976 Louis Nirenberg 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 . 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 , 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 . 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 ; 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 . 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 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 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, , 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 . 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 . 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 , 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

Fefferman. Publications of Hindustan Book Agency are distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for commercial channels.

December 2017 Notices of the AMS 1273 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

D. R. FULKERSON PRIZE

The Fulkerson Prize Committee invites nominations for the Delbert Ray Fulkerson Prize, sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society. Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. The Fulkerson Prize is for outstanding papers in the area of discrete mathe- matics. The prize will be awarded at the 23rd International Symposium on Mathematical Programming to be held in Bordeaux, France on July 1–6, 2018. Eligible papers should represent the final publication of the main result(s) and should have been published in a recognized journal or in a comparable, well-refereed volume intended to publish final publications only, during the six calendar years preceding the year of the Symposium (thus, from January 2012 through December 2017). The prizes will be given for single papers, not series of papers or books, and in the event of joint authorship the prize will be divided. The term “discrete mathematics” is interpreted broadly and is intended to include , networks, mathematical programming, applied combi- natorics, applications of discrete mathematics to computer science, and related subjects. While research work in these areas is usually not far removed from practical applications, the judging of papers will only be based on their mathematical quality and significance. Further information about the Fulkerson Prize can be found at www.mathopt.org/?nav=fulkerson and at www.ams.org/prizes/fulkerson-prize.html.

The Fulkerson Prize Committee consists of Maria Chudnovsky (Princeton University), Friedrich Eisenbrand (EPFL), Chair, and Martin Grötschel (- Brandenburgische Akademie der Wissenschaften). Please send your nominations (including reference to the nominated article and an evaluation of the work) by February 15, 2018, to the chair of the committee. Electronic submissions to Professor Eisenbrand at [email protected] are preferred.

Friedrich Eisenbrand EPFL, Station 8 CALL NOMINATIONS FOR CH-1015 Lausanne [email protected] An Old Problem Resurfaces Nonlocally: Gamow’s Liquid Drops Inspire Today’s Research and Applications

Rustum Choksi, Cyrill B. Muratov, and Ihsan Topaloglu

Gamow’s Liquid Drop Model ABSTRACT. Gamow’s simple liquid drop model of The 1920s may arguably be called the Golden Age of an atomic nucleus recently captured the attention of quantum mechanics. An explosive development of this mathematicians and has inspired numerous advances emerging field of physics drew scores of aspiring re- and open questions in geometric and variational searchers, one of whom was the young Russian theoretical analysis. physicist George Gamow (see Figure 1, page 1276 and sidebar on page 1278.) His name shot to fame when in 1928, at the age of twenty-four, he published a paper in “A mathematician friend of mine, the late S. Banach, which he explained the phenomenon of alpha-decay as once told me, ‘The good mathematicians see analogies a quantum tunneling effect. In the words of , between theorems or theories; the very best see analogies the results of this paper constituted “the first successful between analogies.’ This ability to see analogies between application of quantum theory to nuclear phenomena.” In models for physical theories Gamow possessed to an the paper, Gamow also acknowledged a little help from almost uncanny degree.” Russian mathematician N. Kochin, admitting himself that —Stanislaw Ulam he was “not good in mathematics.” In 1928 Gamow [1] made another important discovery that has become forever linked with his name. During his stay in with in the fall of that year, Rustum Choksi is professor of mathematics at McGill University. His email address is [email protected]. Gamow conceived of what is now known as the liquid drop Cyrill B. Muratov is professor of mathematics at Insti- model of the atomic nucleus. This simple model, which tute of Technology. His email address is [email protected]. was soon refined and further developed by Heisenberg, Ihsan Topaloglu is assistant professor of mathematics at Virginia von Weizsäcker, and Bohr after the discovery of neutrons Commonwealth University. His email address is iatopaloglu in 1932, treats the collection of protons and neutrons @vcu.edu. inside an atomic nucleus as an incompressible, uniformly For permission to reprint this article, please contact: charged fluid. With only a few fitting parameters and an [email protected]. assumption that the nuclei are spherical, this model was DOI: http://dx.doi.org/10.1090/noti1598 able to accurately predict the mass defect curve—the loss

December 2017 Notices of the AMS 1275 where the nucleus Ω ⊂ ℝ3 is a measurable set with fixed volume |Ω| = 푚. We refer to 푚 as “mass,” which is a parameter proportional to the number of nucleons in a nucleus. Per(Ω) is the perimeter of the set Ω, i.e., a suitably generalized notion of the surface measure of 휕Ω.1 The ground state of a nucleus with a given number of nucleons is then the minimizer of 퐸, i.e., the set Ω that achieves the least energy, (2) 푒(푚) ∶= inf {퐸(Ω)∶ |Ω| = 푚}, for a given mass 푚. Ultimately, the purpose of this liquid drop model is to a marriage (or predict 1) the shape of nu- clei, 2) the nonexistence of rather divorce) arbitrarily large nuclei, and of two older 3) the existence of a nucleus with the least energy per nu- geometric cleon (the element having the greatest nuclear bind- problems ing energy). It is precisely the competition between the forces which try to minimize the surface area of the nucleus and those which try to spread the nuclear charges apart that makes answering these questions nontrivial. Gamow’s liquid drop problem is a marriage (or rather divorce) of two older geometric problems: Figure 1. George Gamow (portrait, 1932). At the age of twenty-four, George Gamow introduced the liquid (1) the Classical Isoperimetric Problem of minimizing drop model that successfully explained the basic the perimeter of a body with fixed mass 푚; and characteristics of the atomic nuclei and predicted (2) the Problem of the Equilibrium Figure of a self- nuclear fission as a result of an instability of large gravitating fluid body of mass 푚. spherical nuclei with respect to nonspherical For the first problem, whose roots go back to antiquity, distortions. Schwarz demonstrated the minimizing property of balls in 1884 for piecewise-smooth sets in three dimensions. of mass or energy when nucleons come together to bind in The complete solution was given in 1958 by De Giorgi, the nucleus as a function of the number of nucleons. Thus, who showed that the unique minimizer of the perimeter indirectly, the model also predicted the spherical shape functional among all measurable sets with fixed mass of most nuclei. The model’s ultimate triumph came from is given by a ball. Starting with Newton (1687), the sec- explaining the phenomenon of nuclear fission in terms ond problem attracted the attention of many celebrated of an instability of large spherical nuclei with respect to mathematicians. Assuming zero angular momentum, the nonspherical distortions (Bohr and Wheeler, 1939). The total potential energy of a self-gravitating fluid body, model has also been extensively used in astrophysics represented by a measurable set Ω ⊂ ℝ3, is given, up to a to describe exotic phases of nuclear matter at ultrahigh constant, by densities found in neutron stars (Baym, Bethe, and Pethick, 1 1971). (3) − ∫ ∫ 푑푥 푑푦, |Ω| = 푚, In the modern rendition of Gamow’s liquid drop model Ω Ω |푥 − 푦| the attractive short-range nuclear force gives rise to where −|푥 − 푦|−1 is the potential resulting from the gravi- excess surface energy due to lower nucleon density near tational attraction between two point masses at positions the nucleus boundary, while the presence of positively 푥 and 푦 in the fluid. Lyapunov (1886) made the first charged protons gives rise to a repulsive Coulombic force. mathematical breakthrough by establishing that every Since the Coulomb energy of a proton in a nucleus is regular minimizer of (3) is a ball. Poincaré commented much smaller than its average kinetic energy determined on Lyapunov’s proof in 1887 and went on to make the by strong nuclear forces, to a good approximation the problem famous in his 1902 treatise Figures d’Equilibre spatial distribution of charge in a nucleus is uniform. d’une Masse Fluide. Almost twenty years later, Carleman Therefore, mathematically the energy of a nucleus within (1919) showed that balls are indeed minimizers. Yet it the model may be written (up to shape-independent bulk was not until the work of Lieb (1977) that a complete terms and after a suitable nondimensionalization) as solution based on strict Riesz rearrangement inequality 1 1 (1) 퐸(Ω) ∶= Per(Ω) + ∫ ∫ 푑푥 푑푦, 1 8휋 Ω Ω |푥 − 푦| See “WHAT IS…Perimeter?” in the October 2017 Notices.

1276 Notices of the AMS Volume 64, Number 11 became available. Steiner symmetrization plays a central Although Gamow’s liquid drop model has been very role in the analysis of both problems. well known among physicists, it is surprising that it did not receive much attention in the mathematics community until recently. In 2010 the liquid drop problem resurfaced in an asymptotic study by Choksi and Peletier [2], [3] of the Ohta-Kawasaki functional arising in a completely unrelated field of polymer physics. In fact, it is clear that the liquid drop problem represents a prime example of a problem of pattern formation from energetic compe- titions, which is an area driving much work in modern calculus of variations.

Global Minimizers of the Liquid Drop Problem Global existence for minimizers of 퐸(Ω) with |Ω| = 푚 on unbounded domains is nontrivial because of the lack of compactness of minimizing sequences. What we know so far regarding the problem associated with 푒(푚) can be summarized by the following theorem. Here we consider the minimization problem (2) over all sets of Figure 2. In Gamow’s liquid drop problem, the finite perimeter (Caccioppoli sets). two terms in the energy are in direct competition: Theorem 1. There exist constants 0 < 푚0 ≤ 푚1 ≤ 푚2 the perimeter term favors a single ball, while the such that: Coulomb term favors splitting. For small values of (1) If 푚 ≤ 푚 , then there exists a minimizer. 푚 the perimeter wins completely, whereas for large 1 (2) If 푚 ≤ 푚 , then the unique minimizer is a ball. values of 푚 the Coulomb term dominates, resulting 0 (3) If 푚 > 푚 , then there is no minimizer. in fission. 2 The result in this theorem, in this form, was first Thus, up to translations, the ball of mass 푚 is the provided by Knüpfer and Muratov [6] in 2014. The exis- unique minimizer in the two problems above. Gamow’s tence of solutions for certain values of 푚 was already liquid drop model, on the other hand, puts them in direct noted by Choksi and Peletier [2] in 2010 where the opposition, with balls being best for the perimeter term strict concavity of 푒(푚) for small values of 푚 was also but worst for the Coulomb term. The relative strength of established. Knüpfer and Muratov took a more direct the two terms is controlled by the magnitude of 푚. One approach and proved that for sufficiently small mass can readily see the effect of the size of 푚 as to which every minimizing sequence can be replaced by another term dominates from a simple scaling argument. Namely, minimizing sequence consisting of sets with uniformly by looking at 휆Ω for any 휆 > 0 we see that bounded diameter. This uniform bound provided the necessary compactness for the minimizing sequence, and 휆2 1 휆−3퐸(휆Ω) = 휆−1 Per(Ω) + ∫ ∫ 푑푥푑푦. with the lower semicontinuity of 퐸(Ω) with respect to 8휋 Ω Ω |푥 − 푦| the 퐿1- the existence of minimizers follows from Thus the energy per nucleon 휆−3퐸(휆Ω) is dominated by the direct method of the calculus of variations. They also the perimeter term for small values of 휆 > 0 and by the showed that minimizers of 퐸(Ω) are so-called “quasi- Coulomb term for large values of 휆 > 0 (see Figure 2). minimizers” of the perimeter functional and hence have Therefore, we would be led to expect that the unique smooth boundaries. Implementing the sharp quantitative solution (up to translations) of problem (2) is given by isoperimetric inequality, proved by Fusco, Maggi, and a ball when 푚 is small, whereas this problem does not Pratelli (2008), combined with the regularity, Knüpfer and have a solution when 푚 is large. Also, for fixed Ω there is Muratov proved that balls are, up to translations, unique a unique value of 휆 at which the energy per nucleon (the minimizers of 퐸(Ω) for sufficiently small mass. In order negative of the binding energy) is minimum. to prove that the problem does not have a solution for If one is to believe that Gamow’s liquid drop model large masses, the authors first showed that minimizers correctly captures the ground state configurations of the are connected. However, they also showed that any con- atomic nuclei, then nature tells us that the solution of nected set with sufficiently large mass can be divided into the liquid drop problem is always given by a ball. In two large pieces by a suitable plane. Moving these pieces this case an explicit calculation gives the value of 푚 = far apart from each other, they compared the energy of 40휋 1/3 −1/3 the transformed disconnected set with the original con- 3 (2 + 2 − 1) ≈ 44.1, beyond which minimizers no longer exist. This critical value is such that one ball figuration and concluded the proof via a contradiction of mass 푚 has exactly the same energy as two balls with argument. 1 mass 2 푚 infinitely far apart (Frank and Lieb, 2015). Under the same assumption, the minimum energy per unit mass is attained for 푚 = 10휋 ≈ 31.4.

December 2017 Notices of the AMS 1277 George Gamow George Gamow (1904–1968) was a Russian-American theoretical physicist who introduced some of the most influential ideas to twentieth-century science. Born in 1904 in Odessa, , Gamow graduated from Leningrad University in 1926, where together with fellow students L. Landau and D. Ivanenko he founded the group of the Three Musketeers to discuss the latest topics in theoretical physics, in addition to fooling around with friends and generally having a good time. In the summer of 1928, Gamow undertook his first trip abroad on a four-month fellowship to the University of Göttingen, where he met many of the pioneers of quantum theory. Importantly, the trip also gave Gamow an opportunity to visit N. Bohr in Copenhagen and E. Rutherford at Cambridge, which was instrumental for the development of his quantum mechanical ideas in connection with nuclear phenomena. I shall never forget the first time he appeared in Göttingen—how could anyone who has ever met Gamow forget his first meeting with him— a Slav giant, fair-haired and speaking a very picturesque German; in fact he was picturesque in everything, even in his physics. —L. Rosenfeld Gamow returned to the as a sort of celebrity. Yet the political atmosphere in the country had been rapidly changing, and Gamow felt it personally when in 1931 he was denied an exit visa to go to an international nuclear physics congress in . From that time on, Gamow became obsessed with the desire to escape the confines of the Soviet Union, which in his mind had begun to interfere with the free movement of people and ideas across state borders. After an unsuccessful attempt to cross the Black Sea from Crimea into Turkey with his wife on a kayak, Gamow finally managed to obtain permission to leave the Soviet Union in 1933 for a conference, a trip from which he never returned.2 In 1934, Gamow settled at George Washington University in the United States, where he spent the next twenty-two years, the most productive period of his scientific career. He then moved to the University of Colorado, Boulder, where he worked until his death in 1968. Gamow never received a Nobel Prize in physics, although his greatest achievements included the Big Bang theory, the theory of stellar structure and evolution, and a key insight into the nature of the genetic code. To the general public, however, Gamow is best known as the author of a popular science series, The Adventures of Mr. Tompkins, explaining the fundamental concepts of modern physics to millions of readers.a

aThis issue’s BookShelf (page 1327) features Gamow’s popular text, One Two Three…Infinity: Facts and Speculations of Science.

Julin (2014) independently proved that for 푚 suffi- they obtain an upper bound of the form ciently small the ball of mass 푚 is the unique minimizer for 푒(푚), up to translations. His approach is to use 푒(푚) ≤ 퐶푚, 푚 > 푚0. a stronger version of the quantitative isoperimetric in- equality. This version measures the difference between In order to refine the lower bound of this scaling estimate, perimeters of a set and a ball of the same mass in terms of the oscillations of the boundary quantified by the Lu and Otto prove a density estimate which states that a 퐿2-norm of the difference of generalized outward normal minimizer of 퐸(Ω) cannot be thinner than order 1 in 푚, vectors. and combining this with the connectedness of minimizers, Independently, Lu and Otto (2014) also proved that for they obtain that for large values of 푚, sufficiently large 푚, the energy 퐸(Ω) does not admit a minimizer. By the subadditivity of 푒(푚), 1 ∫ ∫ 푑푥 푑푦 ≥ 퐶 푚 log 푚. 푒(푚) ≤ 푒(푚′) + 푒(푚 − 푚′), 0 < 푚′ < 푚, Ω Ω |푥 − 푦|

2The other members of the Three Musketeers group did not fare Together with the upper bound this estimate yields a so well: L. Landau was arrested in 1938 and nearly died after contradiction when 푚 is sufficiently large. More recently, one year in detention; D. Ivanenko was sentenced in 1935 to a via a very simple averaging technique applied to a cutting three-year term in labor camps, substituted with a two-year exile argument, Frank, Killip, and Nam (2016) proved that if in Tomsk after serving one year in Gulag; M. Bronshtein, another key member of the Leningrad group, was executed by NKVD in 푚 > 32휋 ≈ 100.53, then the problem does not admit a 1938. minimizer.

1278 Notices of the AMS Volume 64, Number 11 The Liquid Drop Model on the Torus and the Ohta-Kawasaki Functional 3 In the astrophysics context, one often considers the liquid drop model on the three-dimensional flat torus 핋ℓ of 3 sidelength ℓ. Specifically, for Ω ⊂ 핋ℓ with |Ω| = 푚 one considers the energy

3 1 −1 (4) 퐸ℓ(Ω) ∶= Per(Ω, 핋ℓ) + ∫ (휒Ω − 휌)(−Δ) (휒Ω − 휌)푑푥, 2 3 핋ℓ 3 3 3 where Per(Ω, 핋ℓ) denotes the perimeter of Ω relative to 핋ℓ, 휒Ω is the characteristic function of Ω, and 휌 = 푚/ℓ is the density of the neutralizing background charge supplied by electrons. While this problem is well known in the studies of the structure of nuclear matter in neutron stars, it only recently received attention in the mathematics literature as the sharp interface version of the Ohta-Kawasaki functional

휀 2 1 2 2 1 −1 1 (5) ℰℓ(푢) ∶= ∫ ( |∇푢| + (1 − 푢 ) + (푢 − ̄푢)(−Δ) (푢 − ̄푢))푑푥, ∫ 푢푑푥 = ̄푢, 3 2 4휀 2 ℓ3 3 핋ℓ 핋ℓ which is a diffuse interface version of the energy in (4) with the asymmetry parameter ̄푢∈ (−1, 1) fixed and 휀 ≪ 1. The energy in (5) was introduced in 1986 by Ohta and Kawasaki as a simple model for self-assembly of diblock copolymer melts, even if its nuclear physics analog goes back all the way to the classical 1935 paper of von Weizsäcker. It can also be considered as a tool for numerical studies of (4). Unlike the liquid drop problem posed on all space, the energy 퐸ℓ always admits a minimizer, and a question closely related to the liquid drop problem is what happens to those minimizers when ℓ → ∞ and/or 푚 → 0. Here the finiteness of the domain forces minimizers to always balance the effects of the perimeter and the Coulomb terms. In fact, numerically the putative global minimizers of the Ohta-Kawasaki functional appear to be periodic, with the interfaces within each intrinsic periodic cell resembling a constant mean curvature surface; e.g., a lamella, sphere, cylinder, double-gyroid, etc., as in Figure 3. Thus numerics suggest a separation of the effects of the perimeter and the nonlocal term: the latter sets an intrinsic periodicity, while the former dictates the interface structure within the period cell. This separation has been exploited in a number of papers by Ren and Wei (2007, 2008, 2009) and by Cristoferi (2016) for constructing critical points of variants of (4) perturbatively. To gain further insights into this separation, an asymptotic analysis was presented by Choksi and Peletier [2], [3] and by Knüpfer, Muratov, and Novaga [4] for the droplet regimes wherein 휌 → 0. Here, Gamow’s liquid drop model (on all ℝ3) emerged as the leading-order energy describing the shape of a single droplet, ignoring the next-order effects of the neighboring droplets.

Figure 3. Level sets of the computed local minimizers of Ohta-Kawasaki energy ℰℓ in (5) as the value of ̄푢 is varied.

While one may conjecture that it is possible to choose They establish this by proving Lipschitz continuity of 푚0 = 푚1 = 푚2, based on physical evidence, to date it 푒(푚) on compact subsets of (0, ∞) combined with the remains open to prove or disprove whether any (or all) of 퐵푉-compactness of sequences of generalized minimizers. the constants 푚푖, 푖 = 0, 1, 2, may be pairwise equal. One Here a generalized minimizer of 퐸(Ω) refers to a finite result in that direction was obtained by Knüpfer, Muratov, collection of sets of masses 푚푗 adding up to 푚, where and Novaga [4] in 2016, who show that the set each set is a minimizer for 푒(푚푗). Around the same time, ℐ = {푚 > 0∶ 푒(푚) is attained} Frank and Lieb (2015) obtained the same result, using a is a closed (hence compact) subset of (0, ∞). diameter bound of the form diam Ω ≤ 퐶푚 for minimizers of 퐸(Ω), along with a concentration-compactness lemma,

December 2017 Notices of the AMS 1279 were carried out by several authors, but, surprisingly, only one well-converged numerical study, by Strutinsky, Lyashchenko, and Popov (1963), appears to exist to date; see Figure 4. Although mathematically it is not in the literature whether suitable nonspherical distortions of balls are Figure 4. Dumbbell-shaped critical points of the critical points, using Lyapunov-Schmidt type reduction liquid drop model for several values of the fissility methods Ren and Wei (2011, 2014) obtained existence parameter 훼 = 푚/(40휋) obtained from the numerical of solutions to the Euler-Lagrange equation which are solution of (6) by Strutinsky, Lyashchenko, and Popov almost of the shape of single and double tori. However, (1963): a three-dimensional schematic (a), profile these more exotic patterns, as shown by Ren and Wei cross-sections in the first quadrant (b). (2017), are highly unstable in the sense of the second variation of the energy. Earlier, oval-shaped solutions in a two-dimensional version of the problem and spherical particularly well-suited to the liquid drop problem, for shell solutions in three dimensions were constructed by a sequence of sets of uniformly bounded perimeter. Ren and Wei (2009), with some numerical studies of Together with strict concavity of 푒(푚) for 푚 ≤ 4휋, these spherical shell and toroidal solutions previously carried results also yield the existence of a minimizer in this out by Wong (1973). range of masses. Obtaining quantitative information about (local) mini- Finally, referring back to the third purpose of the liquid mizers through stability is the approach taken by Bonacini drop model, existence of the tightest bound nucleus was established in 2016 by Knüpfer, Muratov, and Novaga [4], and Cristoferi (2014), based on the explicit calculations of who show that there exists 푚∗ ∈ ℐ such that the second variation of 퐸(Ω) obtained by Muratov (2002) and Choksi and Sternberg (2007). They generalize the 푒(푚) 푓∗ ∶= inf 푓(푚) = 푓(푚∗), 푓(푚) ∶= . result of Acerbi, Fusco, and Morini (2013) for bounded 푚∈ℐ 푚 domains to the unbounded setting of the liquid drop The proof of this fact follows from Lipschitz continuity of problem and prove that strictly stable critical points, i.e., the energy-per-particle 푓(푚) on the set [푚0, ∞) combined sets for which the first variation of 퐸(Ω) vanishes and with compactness of the set ℐ. In fact, the analysis further the second variation of 퐸(Ω) is strictly positive-definite, ∗ shows that the quantity 푓 appears as the leading order are local minimizers. That is, these patterns minimize term in the asymptotic behavior of uniform energy density 퐸(Ω) with respect to competitors close in the 퐿1-topology. of minimizers of 퐸ℓ(Ω) as mass 푚 → 0 and domain size Applying this result to the ball of mass 푚, they conclude ℓ → ∞. Frank and Lieb (2015) also prove the same that the ball is an isolated local minimizer if 푚 < 40휋. existence result for 푓(푚) by relaxing the mass constraint Finally, very recently Julin (2016) proved that every local |Ω| = 푚 and considering inf (퐸(Ω)/|Ω|). 0<|Ω|≤푚 minimizer with sufficiently small mass and perimeter is a ball, using the characterization of compact almost Critical Points and Local Minimizers of the Liquid constant mean curvature surfaces by Ciraolo and Maggi Drop Problem (2015). One of the main physical motivations in the studies of At the heart of these criticality and stability re- Gamow’s liquid drop model was to compute the minimal sults lies the regularity of the boundaries of local energy required for nuclear fission. Bohr and Wheeler minimizers. As was first proved by Sternberg and (1939), following the suggestion of Meitner and Frisch Topaloglu (2011), local minimizers of 퐸ℓ(Ω) have (1939), were the first to analyze the critical points of the 퐶3,훼 boundaries by using classical geometric measure energy in (1). In particular, they wrote down the associated theoretic arguments from reg- Euler-Lagrange equation ularity of minimal surfaces. 1 1 The simplicity (6) 2ℋ(푥) + ∫ 푑푦 = 휆, More recently, Julin and Pisante 4휋 Ω |푥 − 푦| of Gamow’s (2015) demonstrated that the where ℋ(푥) denotes the mean curvature of 휕Ω (positive boundaries of local minimiz- ∞ if Ω is convex) and 휆 is a Lagrange multiplier. Clearly, a liquid drop ers are in fact of class 퐶 , ball of mass 푚 is always a solution of (6). However, as was and Julin (2016) established an- shown numerically in a series of papers from the 1940s model alyticity of the boundary of through the 1970s, many types of nonspherical solutions regular critical points. Since to (6) are possible, including ovoid, dumbbell, pear- conceals the the regularity arguments are shaped, and toroid nuclei. In fact, when the ball of mass true richness. local arguments, they immedi- 푚 passes the stability threshold of 푚 = 40휋 ≈ 125.66, it ately apply to local (and global) bifurcates into a family of spheroids, first identified by minimizers of the liquid drop Bohr and Wheeler (1939), using formal bifurcation analysis problem on the whole of ℝ푛. Furthermore, any critical near the distortion instability. Numerical continuation point of the energy in the 퐿1-topology is what is known as studies of those solutions into the subcritical mass region a Λ-minimizer of the perimeter and hence is also regular.

1280 Notices of the AMS Volume 64, Number 11 ℝ푛 but replace the Newtonian potential with a general Riesz kernel with 0 < 훼 < 푛. In fact, the case 푛 = 2 and 훼 = 1 arises in the physical modeling of charge conden- sation in high-푇푐 superconductors. In this case existence of minimizers with small masses was again obtained by Knüpfer and Muratov [5], [6], where minimality of balls of small masses was shown for 푛 = 2 and for all 3 ≤ 푛 ≤ 7 Figure 5. Examples of critical points of the liquid and 훼 < 푛 − 1. The latter result was extended to 푛 ≥ 8 drop model on the sphere. In the first two panels, by Bonacini and Cristoferi (2014). The case of 푛 ≥ 3 axisymmetric configurations with three and four and 푛 − 1 ≤ 훼 < 푛 was settled by Figalli, Fusco, Maggi, interfaces are shown. The third panel is the result of Millot, and Morini (2015), who also considered a nonlocal a hybrid numerical simulation by Shahriari, Ruuth, generalization of the perimeter functional to fractional and Choksi, where a spiraling pattern achieves a perimeter. We note that the latter has been the subject of stable low energy state. considerable attention recently, starting with the work of Caffarelli, Roquejoffre, and Savin (2010), and gives rise to nonlocal minimal surfaces. In contrast with Gamow’s Extensions and Related Problems model, however, the fractional perimeter represents a The simplicity of Gamow’s liquid drop model conceals nonlocal generalization of the attractive potential that the true richness of behaviors exhibited by the solutions keeps the “liquid drop” together, and, therefore, it alone of this geometric variational problem. At the same time, does not produce the phenomenology of Gamow’s liquid it is clear that the main ingredients of Gamow’s model, drop problem. Back to the liquid drop problem with a namely, the competition between the short-range attrac- Riesz kernel, a nonexistence result for large masses is tive and long-range repulsive forces, are not unique to available for 훼 < 2 (Knüpfer and Muratov [5], [6]). Again, that model and should be expected to produce interesting it is an open problem whether or not large nonspherical behaviors in a variety of contexts, both mathematically minimizers could exist for 푛 ≥ 3 and 훼 ≥ 2. Lastly, in the and physically. special case 푛 = 2 and 훼 sufficiently small, a complete The simplest natural mathematical generalization of solution of the liquid drop problem with a Riesz potential Gamow’s liquid drop model is to extend it to other space was given by Knüpfer and Muratov [5] in 2013. The ob- 푛 dimensions. Considering Ω ⊂ ℝ with 푛 > 3 amounts tained solution agrees with the physical conjecture about to replacing the Newtonian kernel in (1) with a Riesz the structure of the minimizers of the classical Gamow’s −훼 kernel |푥 − 푦| , where 훼 = 푛 − 2. For this problem, liquid drop problem mentioned earlier. it was shown by Knüpfer and Muratov [5], [6] that Moving on to further extensions, we note that Coulom- minimizers still exist for all sufficiently small masses bic repulsive forces naturally arise in a number of other and that minimizers for yet smaller masses are balls physical contexts. Perhaps the best known example goes when 푛 ≤ 7. Julin (2014) extended this result to all back to Lord Rayleigh (1882), who considered equilibrium dimensions by removing the reliance on the classical shapes of charged liquid drops that are perfect conduc- regularity of minimal surfaces. Yet, the arguments leading tors. The associated energy functional may be written in to nonexistence no longer apply, and it is an open problem a nondimensional form as whether or not minimizers to the liquid drop problems, 휆 푑휇(푥)푑휇(푦) which must then be nonspherical as it is always better (7) 퐸(Ω) ∶= Per(Ω) + inf ∫ ∫ , 8휋 휇∈풫(Ω) |푥 − 푦| to split one ball into two identical balls far apart for Ω Ω 푚 ≫ 1, exist for arbitrarily large masses when 푛 ≥ 4. where Ω ⊂ ℝ3 with |Ω| = 푚, as before, and the infimum is The absence of such a nonexistence result also prevents taken over all probability measures supported on Ω. One proving existence of minimizers with optimal energy per would naturally be led to believe that the minimization unit mass. problem for (7) would behave quite similarly to the On the other hand, Gamow’s liquid drop problem be- Gamow liquid drop problem. Yet, Goldman, Novaga, and comes trivial for 푛 < 3, since in that case the fundamental Ruffini (2015) showed that, surprisingly, the minimization solution of Laplace’s equation is no longer positive, and problem for (7) does not admit minimizers for any 푚 > 0 minimizers of the liquid drop problem do not exist for and 휆 > 0. Even worse, the problem does not exhibit any value of 푚 > 0. The problem is still meaningful local minimizers in any reasonable sense, indicating that if posed instead on a compact Riemannian manifold, the energy in (7) does not provide a complete physical and results for global minimizers on a two-dimensional picture as a model of charged liquid drops, as shown by flat torus or a two-dimensional sphere are available; see Muratov and Novaga (2016). The reason for this behavior Sternberg and Topaloglu (2011) and Topaloglu (2013). is a kind of incompatibility between the sets seen by the Locally minimizing axisymmetric critical patterns on the perimeter, which have finite two-dimensional Hausdorff sphere as those in Figure 5 have also been investigated measure of the measure theoretic boundary, and sets by Choksi, Topaloglu, and Tsogtgerel (2015). The one- of positive capacity, which may have positive Hausdorff dimensional periodic case is considerably simpler and measure of dimension 1 < 푑 < 2. This phenomenon also was solved completely by Ren and Wei (2003). Another gives rise to singular solutions of the associated Euler– alternative is to consider the problem on the whole of Lagrange equation, such as the famous Taylor cone with

December 2017 Notices of the AMS 1281 Figure 6. A sequence of snapshots of a levitating charged liquid drop reaching the threshold of the charge-driven instability as the liquid in the drop evaporates. an opening half-angle of ≈ 49.3∘, obtained by Taylor model’s phenomenology shared by many other systems (1962). Further regularity assumptions on the set, such as operating at very different length scales: from femtometer convexity of Ω, yield existence, as was shown by Goldman, nuclear scale to nanoscale in condensed matter systems, Novaga, and Ruffini (2016). It is also clear that the model to centimeter scale for fluids and autocatalytic reaction- in (7) is quite sensitive to the space dimensionality. In fact, a complete solution of a variant of the minimization diffusion systems, all the way to cosmological scales. It problem for (7) in which the set Ω is flat, i.e., Ω = 퐷 × {0}, has generated much recent attention by mathematicians where 퐷 ⊂ ℝ2, with |퐷| = 푚, is a closed set and the and continues to this day to entice and challenge us with perimeter is replaced by the one-dimensional Hausdorff its beguilingly simple yet rich structure. measure of 휕퐷, has been obtained by Muratov, Novaga, and Ruffini (2017), who prove that the minimizer with References “mass” 푚 and “charge” √휆 exists if and only if 휆 ≤ 4푚/휋 [1] G. Gamow, Mass defect curve and nuclear constitution, Proc. and is given by a disk. Note that here a single disk remains R. Soc. Lond. A 126 (1930), no. 803, pp. 632–644. a minimizer against splitting into two equal disks for all [2] R. Choksi and M. A. Peletier, Small volume fraction limit 휆 < 4푚√2/휋. Thus, the scenario of existence failure is of the diblock copolymer problem: I., Sharp-interface func- different from the one hypothesized for Gamow’s liquid tional, SIAM J. Math. Anal. 42 (2010), no. 3, pp. 1334–1370. drop model: instead of splitting into two disks, a single MR 2653253 (2011j:49092) disk breaks into one big disk with a certain amount [3] , Small volume-fraction limit of the diblock copolymer problem: II., Diffuse-interface functional, SIAM J. Math. Anal. of charge and many tiny disks that carry the excess 43 (2011), no. 2, pp. 739–763. MR 2784874 (2012e:49097) charge to infinity to reduce energy for 휆 > 4푚/휋. We [4] H. Knüpfer, C. B. Muratov, and M. Novaga, Low density note that in physical experiments the failure of existence phases in a uniformly charged liquid, Comm. Math. Phys. 345 of minimizers in the three-dimensional charged liquid (2016), no. 1, pp. 141–183. MR 3509012 drop model manifests itself spectacularly via thin jets [5] H. Knüpfer and C. B. Muratov, On an isoperimetric prob- of electrified liquid emitted from the droplet as its lem with a competing nonlocal term I: The planar case, smoothness is lost, as in Figure 6. Smoothness is then Comm. Pure Appl. Math. 66 (2015), no. 7, pp. 1129–1162. restored as a significant portion of charge escapes the MR 3055587 [6] , On an isoperimetric problem with a competing non- drop via the jet. local term II: The general case, Comm. Pure Appl. Math., 67 (2014), no. 12, pp. 1974–1994. MR 3272365 Conclusion Gamow’s liquid drop model is indeed a very simple Image Credits model for a highly complex system. However, its beauty Figure 1 (portrait of George Gamow, 1932) courtesy of Elfriede is really in tune with the famous quote attributed to and Igor Gamow. Albert Einstein: Everything should be made as simple as Figure 2 created by the authors. possible, but not simpler. While it was initially posed to Hand-drawn portrait of George Gamow courtesy of A. D. Chernin, describe nuclear structures, the fact that it encapsulates “Gamow in America: 1934–1968 (On the ninetieth anniver- a rather ubiquitous competition of short- and long-range sary of G. A. Gamow’s birth),” Physics–Uspekhi, vol. 37, effects is behind a universality, with the liquid drop pp. 791–802, 1994, reprinted with permission.

1282 Notices of the AMS Volume 64, Number 11 AMERICAN MATHEMATICAL SOCIETY

Figure 3 simulations by J. F. Williams, discussed in Choksi, Peletier, and Williams (2009). Figure 4a created by the authors. Figure 4b reprinted from Nuclear Physics 46, V. M. Strutinsky, N. Ya. Lyashchenko, and N. A. Popov, “Symmetrical shapes of equilibrium for a liquid drop model,” pp. 639–659, July– September 1963, with permission from Elsevier. Figure 5 courtesy of B. Shahriari, R. Choksi, I. Topaloglu, and mathematics G. Tsogtgerel, “Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere,” ESAIM: Con- trol, Optimisation and Calculus of Variations, vol. 21, no. 1, LANGUAGE OF THE SCIENCES pp. 247–270, 2015, reprinted with permission. Figure 6 courtesy of E. Giglio, B. Gervais, J. Rangama, B. Manil, B. A. Huber, D. Duft, R. Müller, T. Leisner, and C. Guet, “Shape deformations of surface-charged microdroplets,” Physical Review E 77, 036319, 2008. Copyright 2008 by the Ameri- can Physical Society. dx.doi.org/10.1103/PhysRevE.77 .036319. Reprinted with permission.

ABOUT THE AUTHORS Rustum Choksi focuses his math- ematical research on the calculus of variations and its applications to pattern formation. Since he was a child, he has been a passion- ate lover of classical music, from chamber music to opera. He also devotes too much time and money Rustum Choksi to collecting minerals, in partic- ular agates with their alluring and infinite-dimensional pattern engineering morphology. astronomy robotics Cyrill B. Muratov is interested in genetics the emergence of spatial complex- medicine ity from basic constitutive laws. biology He uses a combination of rigor- climatology ous , formal forensics asymptotics, and numerical sim- statistics ulations to understand complex fi nance physical phenomena arising in computer science science and engineering. physics Cyrill B. Muratov neuroscience chemistry Since high school, Ihsan Topaloglu geology has been fascinated by the con- biochemistry nection between mathematics and physics. He now studies pattern ecology formation in models of materi- molecular biology als and biological aggregations through geometric variational problems.

Ihsan Topaloglu

December 2017 Notices of the AMS 1283 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

Associate Editor, Mathematical Reviews Algebra | 2018

Applications are invited for a full-time position as an Associate Editor of Mathematical Reviews/MathSciNet®, to commence as early as possible in late spring/early summer 2018. We also welcome applications from suitably qualified candidates interested in spending a sabbatical year from their institution in the position and continuing as an editor working remotely under contract afterwards. The Mathematical Reviews (MR) division of the American Mathematical Society (AMS) is located in Ann Arbor, Michigan, in a beautiful, historic building close to the campus of the . The editors are employees of the AMS; they also enjoy certain privileges at the university. At present, the AMS employs approximately eighty people at Mathematical Reviews, including sixteen mathematical editors. MR’s mission is to develop and maintain the MR Database, from which MathSciNet is produced. An Associate Editor is responsible for broad areas of the mathematical sciences. Editors select articles and books for coverage, classify these items, determine the type of coverage, assign selected items for review to reviewers, and edit the reviews on their return. The successful applicant will have mathematical breadth with an interest in current developments, and will be willing to learn new topics in pure and applied mathematics. In particular, we are looking for an applicant with expertise in algebra and an interest in a range of algebraic topics, such as representation theory, nonassociative algebras, and group theory. The ability to write well in English is essential. The applicant should normally have several years of relevant academic (or equivalent) experience beyond the PhD Evidence of written scholarship in mathematics is expected. The twelve-month salary will be commensurate with the experience that the applicant brings to the position. The review of the applications will begin on February 12, 2018 and will continue until the position is filled. Applications can be submitted through MathJobs at www.mathjobs.org/jobs/jobs/10667 Inquiries may be sent to the Executive Editor: Edward Dunne Executive Editor, Mathematical Reviews American Mathematical Society 416 4th Street Ann Arbor, MI 48103 USA [email protected] The American Mathematical Society is an Affirmative Action/Equal Opportunity Employer. Marcel Berger Remembered

Claude LeBrun, Editor and Translator

Gérard Besson EDITOR’S NOTE. Claude LeBrun has kindly assembled this memorial for Marcel Berger. This month also Marcel Berger,1 one of the world’s leading differential marks the conference “Riemannian Geometry Past, geometers and a corresponding member of the French Present and Future: An Homage to Marcel Berger” at Academy of Sciences for half a century, passed away on IHÉS. October 15, 2016, at the age of eighty-nine. Marcel Berger’s contributions to geometry were both broad and deep. The classification of Riemannian holo- nomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to alge- braic geometry. His 1960 proof that a complete oriented even-dimensional manifold with strictly quarter-pinched positive curvature must be a topological sphere is the Claude LeBrun is professor of mathematics at Stony Brook Univer- sity. His email address is [email protected]. Gérard Besson is CNRS Research Director at Institut Fourier, Uni- For permission to reprint this article, please contact: versité Grenoble, France. His email address is g.besson@ujf [email protected]. -grenoble.fr. DOI: http://dx.doi.org/10.1090/noti1605 1Pronounced bare-ZHAY.

December 2017 Notices of the AMS 1285 research director by the CNRS. He served as president of the French Mathematical Society (SMF) during the period of 1979–80 and in this capacity helped oversee the foundation of the International Center for Mathematical Workshops (CIRM) in Luminy. In 1985 he then became director of the Institute for Higher Scientific Studies (IHÉS) in Bures-sur-Yvette, a position in which he served until 1993, when he was succeeded by his former student Jean-Pierre Bourguignon. The mathematical community will remain eternally indebted to Marcel Berger for his marvelous contributions to our mathematical heritage, which have substantially transformed the field of differential geometry. Misha Gromov Marcel Berger, during his directorship (1985–1994) of the IHÉS, in Bures-sur-Yvette, France. Inspired by Marcel Berger: Recollections (1955–1995) There is no need for me to say anything direct ancestor of a vast sector of subsequent research in about Marcel’s famous 1/4-pinching theorem, his global Riemannian geometry. Through his many students discovery of collapse with bounded curvature, and collaborators, he created a school which carried the his contributions and leadership in the field of torch of differential geometry into a new era. Under his spectral geometry, or his classification of special leadership, a group of mathematicians collaborating un- holonomy groups—everybody in our universe3 is der the pseudonym of Arthur L. Besse produced several aware of these. landmark books [1], [19], [20] that inspired a ferment of research activity in connection with key topics of interest: However, it is perhaps worth mentioning that the the study of Riemannian manifolds whose geodesics are last was cited as a source of inspiration by Jim all closed, links between volume and injectivity radius Simons when the Légion d’Honneur was bestowed [11], and the theory of Einstein manifolds [10]. He has on him by the French ambassador in New York. also left us an impressive series of popular and peda- (1972) Systolic Inequalities and Calibrations. In a couple gogical books, as exemplified by his massive treatise A of articles from this period, Marcel proposed a new Panoramic View of Riemannian Geometry [14]. His efforts conceptual (systolic) rendition of Loewner’s early in this last vein also included a campaign to increase the results on surfaces. With characteristic modesty, understanding of the work of Mikhail Gromov. he attributed the idea to René Thom. Born in Paris on April 14, 1927, Berger was a student at the École Normale Supérieure from 1948 until 1953, when In his evaluation of systoles of particular man- he was awarded a research assistantship by the CNRS2 ifolds, I believe Marcel made the very first use after distinguishing himself in a national competition. ever, in the context of quaternionic spaces, of the In 1954, under the supervision of André Lichnerowicz, method of calibrations and proved that he then defended his doctoral thesis, which proved The quaternionic projective subspaces ℍℙ푘 ⊂ ℍℙ푛 a landmark classification of the holonomy groups of are volume minimizing among all 4푘-cycles not Riemannian manifolds. homologous to zero in ℍℙ푛. After spending the academic year 1956–57 at MIT, Berger was promoted to the rank of research scientist by (1978) Blaschke Conjecture for Spheres. If a Riemann- the CNRS. In 1958 he then accepted a junior appointment ian manifold 푋 homeomorphic to 푆푛 has diameter at the University of Strasbourg, where he was eventually equal to its injectivity radius, then 푋 is isometric to promoted to full professor in 1962, shortly after his return a standard “round” (constant-curvature) sphere. from a year in Berkeley. After moving to a professorship Marcel’s dazzlingly simple proof of this (which in Nice for two years, he then moved again in 1966, involves a contribution by Jerry Kazdan, who this time to the University of Paris, which at that time proved a key analytic inequality suggested by was still a single, unified institution. When the university Marcel), hardly takes half a page. was broken into smaller units in the wake of the 1968 student riots, Berger was attached to the University of Paris VII, at Jussieu, and was promoted to the rank of Mikhail Gromov is Jay Gould Professor of Mathematics at the Courant Institute, New York University, and professor emeritus at 2The Centre national de la recherche scientifique is the French Institut des Hautes Études Scientifiques, France. His email address government’s main agency for the support of fundamental sci- is [email protected]. entific research. Unlike the American NSF, the CNRS supports a 3Here, “our universe” means the “community of differential large network of scientists on a permanent basis. geometers.”

1286 Notices of the AMS Volume 64, Number 11 (1980) Isoembolic Inequality. Among all 푛-dimensional My thesis in 1961 was a direct result of his remarkable manifolds with a given injectivity radius, the round work on holonomy in irreducible Riemannian manifolds, sphere has the least volume. in which he enumerated all possible holonomy groups. These all turned out to be transitive on the unit sphere, (2000?) Buying Cheese in Paris. One day, while a guest which led to the question of how one could prove this at Marcel’s apartment in Paris, I tried a piece of directly without recourse to his list. Such a proof was cheese whose flavor far excelled anything I had provided by my thesis. ever tasted before. When I asked Marcel where this Berger also proved the first cheese came from, he could not hide his surprise pinching theorems in Riemann- at the impudence of my question. ian geometry, proving that This was Of course—he said—it is from Quatrehomme. Do an even-dimensional compact you think we would serve our guests cheese from Riemannian manifold whose seminal work. Dubois? sectional curvature is strictly 1 Please,—I said—what is the name of this cheese, between 4 and 1 is necessarily homeomorphic to the and where is this boutique? I want to go there and sphere and later showing that, if we don’t insist that buy this cheese myself. the inequalities be strict, the only exceptions we’ve now allowed in are just the symmetric spaces of rank 1. This Marcel literally froze, with his mouth open. was seminal work. I showed it to my first student, Jeff Yourself? Are you kidding? Do you imagine that Cheeger, and it inspired him to write an excellent thesis. you will just go to Quatrehomme and buy Pont- Jeff then went on to develop these ideas further, initially l’Évêque??!! alone and later in collaboration with Detlef Gromoll. This illustrates the way that Berger’s ideas quickly led to an im- Why not? What could be complicated about buying portant body of work, relating curvature to the topology cheese? of Riemannian manifolds. Of course, it is simple. But getting good cheese takes time—about a year, maybe, a half-year if Jeff Cheeger you really know how to go about it. Marcel Berger was one of the leading Riemannian geome- By this time, it was my own mouth that was ters of the mid-twentiethth century and for many years agape. Taking pity on my ignorance, Marcel then the dean of French differential geometry. His beautiful explained, contributions to metric geometry, especially those from A respectable cheese has a narrow time window 1955 to 1980, provided a high standard for the period of 3–4 days during which it is good to eat. Only and played a major role in setting the stage for the the proprietor of a boutique knows which of his subject’s next explosive phase of development. A few of cheeses are ripe. If you are a serious customer, he the highlights were the even-dimensional sphere theorem will tell you which cheeses you should buy on a with (sharp) 1/4-pinching, associated results such as the particular day. minimal diameter rigidity theorem (arguably, the first global rigidity theorem), the sphere theorem with pinch- To gain his respect you have to come to the shop ing slightly below 1/4 (via a compactness/contradiction regularly, at least once a week, and buy a couple argument), and his comparison theorem, known as “Rauch kilograms of several cheeses. Besides—and this is II”; for details, see [4], [6], [12], [7]. crucial—you should show some understanding of With hindsight, Berger’s lemma, concerning pairs of cheese. points at maximum distance, to some extent foreshad- Eventually, he will accept you, and then you will owed the Grove–Shiohama–Petersen theory of critical start getting cheeses like the one you have just points of distance functions, which has had a number of tasted. very remarkable applications. Ourselves, concluded Marcel— Berger’s example of a collapsing sequence of positively curved metrics on 푆3 is now viewed as the first nontrivial We do it together. Five families! One of us, always example of collapse with bounded curvature. It is very the same man, has gone to Quatrehomme for five illuminating and remains required reading for anyone years. Simple, and it works! interested in learning the theory of collapse with bounded curvature. To fully appreciate its significance, one must Jim Simons be aware that at the time, Riemannian geometry was very short of “meaningful” examples. The point for Berger I very much admired Marcel Berger. He was an exceptional was that it demonstrated that the known lower bound mathematician and a gracious individual. His death is a estimates on the injectivity radius in the odd-dimensional sad loss to the community. simply connected case no longer held for pinching < 1/9.

Jim Simons is chairman at Renaissance Technologies and research Jeff Cheeger is Silver Professor of Mathematics at Courant Institute, professor of mathematics at . New York University. His email address is [email protected].

December 2017 Notices of the AMS 1287 all that much. Analysis was not playing the central role that it came to play after the revolutionary impact of the work of Yau. That was just as well, since at the time, I didn’t know much analysis. During the first few years after my degree, almost all of my work was in metric geometry, much of it in collaboration with Detlef Gromoll. My book with David Ebin, Comparision Theorems in Riemannian Geometry, contained an exposition of many of the key results up to 1975, including, of course, various theorems of Berger. Berger’s fundamental classification of the candidates for holonomy groups of Riemannian manifolds, a tour de force, was accomplished [2], [3] during the period 1953– 55. An unexplained consequence was the observation that if the holonomy group of 푀푛 acts irreducibly, then either it acts transitively on the unit sphere or 푀푛 is a symmetric space of rank ≥ 2. In his thesis, Jim Simons gave an algebraic proof of this fact, one which avoided the classification. Later, Jim became my teacher and directed me to the study of comparison theorems. He said, “There isn’t too much else going on in geometry right now.” (And this just slightly before his own spectacular work on minimal varieties and Chern–Simons invariants!) Berger visited Berkeley for the “big” Global Analysis Summer Institute, which lasted for three weeks during the summer of 1968. I was just finishing up my postdoc, and at some point I was told that the great man would like to meet with me to discuss geometry. Excited but a bit nervous, I asked if I could bring along my mathematical big brother, Detlef. Naturally, it turned out that Marcel (as I will now call him) was kind and friendly. A few summers later, at Marcel’s invitation, Detlef and I spent several weeks in Paris. He was surrounded by a large and impressive group of students, including Jean-Pierre Berger in Paris in 1968. Bourguignon, with whom we later became good friends. During the visit we asked Marcel if he could recommend Berger’s sharp upper bound for the injectivity radius a restaurant. He sent us to a place with the amusing name in terms of the volume [9], [11] is remarkable in that Le Sanglier Bleu, where we had a great meal and a super good time. curvature does not enter. Closely related work with Jerry It turned out that Marcel had become interested in Kazdan was used in their proof [18] of the Blaschke the spectrum of the Laplacian, especially the question conjecture: Manifolds for which the cut locus of every of what geometric information might be contained in point consists of a single point are isometric to standard the asymptotic expansion of the trace of the heat kernel. spheres. The coefficients in this expansion are local invariants As a graduate student interested in learning Riemann- involving the curvature and successively more of its ian geometry, I was enormously impacted by Berger’s covariant derivatives. Attention had been drawn to them work. It was inspiring. I became excited by the idea that 4 by the visionary paper of McKean and Singer, in which one could make progress in geometry just by drawing they proposed the heat equation method for proving the simple pictures, staring at them, waiting for the light to go Atiyah–Singer index theorem for the operator 푑 + 훿 ∶ on, and at the appropriate point bringing in some ordinary Λ푒푣푒푛 → Λ표푑푑, whose index is the Euler characteristic, differential equations and perhaps some topology. A bit and expressed the hope that a “fantastic cancellation” in later, when I was a postdoc at Berkeley, I was taken aback the coefficients would lead to the Chern–Gauss–Bonnet when Dennis Sullivan said, “Don’t you think that it’s just integrand. a little bit naive?” Something significant which came out of the Laplacian From my personal perspective, Riemannian geometry project was Marcel’s book with Gauduchon and Mazet, Le was virtually a perfect subject. It had received tremendous spectre d’une variété riemannienne [17]. During our visit, impetus from the geometrical tools introduced by Rauch, he told me, “My less strong students, I put them on the Berger, Klingenberg, and Toponogov, and a bit later by Gromoll and Meyer. Yet it was still in a very underdevel- 4Curvature and the eigenvalues of the Laplacian, J. Diff. oped state. One could get started without having to know Geometry 1 (1967), 43–69.

1288 Notices of the AMS Volume 64, Number 11 spectre.” This was puzzling, since it seemed to me that he his natural shyness in order to establish a seminar and didn’t have any “less strong” students. Also, I was vaguely develop strong working bonds with a group of research aware of the Ray–Singer conjecture on the equality of students. One consequence was that he agreed in 1969 to Reidemeister torsion and ana- become my thesis adviser, taking over a role previously lytic torsion, which struck me fulfilled by Gustave Choquet. This turn of events was in as very exciting, though I could an aversion to part made possible by the fact that I had attended his only understand it in a superfi- taking the graduate course on the spectrum of a Riemannian mani- cial way. I knew that I needed fold, notes on which were at that time becoming famous to learn more about the heat credit for his under the acronym BGM (for Berger–Gauduchon–Mazet). kernel, and Le spectre d’une But I came to Riemannian geometry largely by default variété Riemannienne was the own after failing to convince any leading Parisian expert on ideal place to start. I decided to classical mechanics to supervise my project of solving make an intensive study of this mathematical the Euler equations by applying Arnold’s strategy to the book, which worked out very group of diffeomorphisms—a strategy that David Ebin and well. contributions Jerrold Marsden in any case successfully developed soon In the period 1980–2000, I afterwards. This left me to master Riemannian geometry encountered Marcel many times during stays at IHÉS, from scratch, having to learn almost everything. where for most of that time he was the director. As usual, The next year, however, because I was Berger’s only he was nice and friendly, but at some point it struck protégé to currently hold a CNRS attaché de recherche me that his personality had a side which seemed slightly position, Marcel bestowed upon me the amazing gift “quirky,” though not quite in a way on which I could of a significant block of his time each week. Every put my finger. One thing I did eventually realize was Tuesday he would spend the whole day trying to tell that he had an aversion to taking the credit for his own me everything he knew about geometry—and that, of mathematical contributions. course, was vast: curvature and topology, the geometry Marcel wrote a large number of excellent expository of geodesics, holonomy, Kähler geometry, the Calabi works on different aspects of geometry. A number of conjecture, etc. The topics ran the gamut of geometric them were at an “elementary” level, though perhaps from research, and the approach he took was holistic. a somewhat advanced standpoint. Most notable for me Marcel waxed enthusiastic when I suggested that David were the monumental A Panoramic View of Riemannian Ebin might spend a year in Paris teaching us global anal- Geometry [14] and the shorter Riemannian Geometry dur- ysis. He had previously collaborated with David during ing the Second Half of the Twentieth Century [13]. While he a visit to Berkeley, and Marcel now clearly foresaw the was preparing these, we had quite a few email exchanges. remarkably transformative role that nonlinear partial dif- I was impressed by his strong desire to get the history ferential equations would surely play in the development right, no matter how much back and forth it took, and of geometry in the years to come. once again by his aversion to explicitly awarding himself Marcel insisted that his Wednesday seminar (which, the credit for his own work. On several occasions I was per local tradition, also bore his name) should be a forced to insist, “But, Marcel, it was you who did that!” real working seminar, with a year-long theme providing Marcel was an outstanding mathematician and a unique the focus for the lectures. While he would always have character. I miss him. the last word when it came to the choice of topics, he consistently gave us all the opportunity to present our Jean-Pierre Bourguignon most recent results. We learned a huge amount, because Marcel’s ambition was for everyone to become acquainted I am indebted to Marcel Berger for a multitude of things with every topic necessary for a deeper understanding that have decisively influenced my mathematical life. of geometry. He nonetheless gave his students plenty These reflect not only his unique place in the rich land- of leeway to develop their own research programs. For scape of late-twentieth-century French mathematics but also his unique personal qualities: his generosity, his geo- example, he in no way objected to my own decision metric vision, his international network of friendships, his to dedicate my thèse d’État to the development of a modesty, his high standards, and his strong commitment “stratification of the space of Riemannian structures.” to public service. Berger developed a network of friendships with inter- Having given up previous academic positions in Stras- national mathematical leaders and was always eager to bourg and Nice, Marcel Berger became a professor in see his students benefit from this network. His strong Paris just in time to experience the tumult of the 1968 ties with Wilhelm Klingenberg gave us the benefit of student riots. Perceiving that society was undergoing a participating in the “Small Arbeitstagung in Differential transformational shift in relations between generations, Geometry,” which provided us with our first real interna- he concluded that it was imperative that he overcome tional exposure, since, paradoxically enough, Bonn was more international than Paris at the time. Then, in June Since 2014, Jean-Pierre Bourguignon is president of the Euro- 1972, a visit to Paris by Jim Simons, who had re-proved pean Research Council, Brussels, Belgium. His email address is Berger’s classification of Riemannian holonomy from a [email protected]. new viewpoint, afforded me the opportunity to spend

December 2017 Notices of the AMS 1289 the academic year 1972–1973 in Stony Brook, which had become the Mecca of differential geometry, with no fewer than fourteen specialists on its faculty. Spending a year in the US had a decisive impact on my own career; it not only resulted in my first joint work with S. T. Yau, who was there in his first professorial position, but also allowed me to attain a more global perspective on my own work. This visit was followed by a summer spent at Stanford at the invitation of Robert Osserman, again on Berger’s recommendation. These contacts then led to sev- eral other opportunities. For example, participation in the Differential Geometrie im Grossen meetings at Oberwol- fach, organized by S.-S. Chern and Klingenberg, became the source of many unforgettable experiences and gave me privileged access to Chern because of the latter’s con- siderable respect for Berger. The good relations Marcel developed with during a stay at MIT opened other doors for me and led to inspiring discussions of Berger (right) talks with Hermann Karcher at the relationship between geometry and physics. Similarly, Oberwolfach in 1982. Also present: Maung Min-Oo Berger’s close ties with Shingo Murakami led to my first (left) and Shoshichi Kobayashi (middle). visit to Osaka in 1979, followed by many other trips to Japan; Berger’s work is highly influential in Japan and has successful venture would not have been possible without provided the starting point for many papers by leading Berger’s leadership and his generous, open personality. Japanese mathematicians. And the list of course goes Of course, much more could be said. Marcel’s conviction on, as Berger’s international contacts put me in contact that Mikhail Gromov’s ideas would completely transform with such figures as Eugenio Calabi, Manfredo do Carmo, differential geometry was quickly borne out by events, and , Jerry Kazdan, and Shoshichi Kobayashi, to the powerful impact of this new perspective on French name only a few. mathematics became particularly evident after Gromov The “Berger school” eventu- decided to settle at the IHÉS. Berger went on to become a ally chose to cloak itself under tireless and persuasive public advocate of Gromov’s ideas Besse was the collective pseudonym of and vision.7 intended as a Arthur L. Besse after a founda- But here I would really have to enter into yet another tional “Round Table” workshop chapter of Berger’s life, namely, his directorship of the direct in the village of Besse-en- Institut des Hautes Études Scientifiques (IHÉS). Although I Chandesse. Besse’s last name eventually became Berger’s successor there, I am actually challenge to was of course provided by less qualified to discuss this period than certain other the village, while his first and contributors to this collection of reminiscences, and I will Bourbaki. middle names of Arthur and therefore leave it to them to comment on this important Lancelot came from the me- part of Marcel’s life and career. Paradoxically, much dieval legends of the Knights of the Round Table.5 In a of my prior knowledge regarding the running of the very real sense, Besse was intended as a direct challenge institute was gleaned from conversations with Marcel’s to Bourbaki: Besse’s philosophy emphasizes the need predecessor, Nicolaas Kuiper, during the period of his to mobilize many different specialized bodies of knowl- own mandate. Since the duties of the director of the edge to attack deep questions, in complete contrast to IHÉS do not include the selection of his successor, I next the model provided by Bourbaki’s Éléments de mathéma- discussed my views concerning the institute’s future, not tiques. Indeed, the approach adopted by Besse was rather with Marcel, but rather with the institute’s permanent similar to the one previously advocated by André Weil in professors. Only when my term as director was about his Publications de l’Université de Nancago,6 much to the to start and upon my return from an extended stay at discomfiture of some of Weil’s Bourbaki collaborators. In MSRI did Marcel share with me some of his vision and any case, Besse’s approach soon justified itself by bearing experience concerning the institute. tangible fruit. It also became evident that this exciting and Marcel’s absence presents us all with a profound loss. We will miss his frequent exhortations to think 5While nearly every child in the English-speaking world seems to geometrically, as well as his gentle manner and modest have heard tales of King Arthur, Lancelot, and the Knights of the attitude, which will remain a model for us all. Round Table, these characters are perhaps best known in France to those with something of a scholarly interest in the medieval 7Berger’s masterful survey of Gromov’s work, “La géométrie selon French romances that are the main sources of Arthurian legend. Gromov,” a lecture delivered at the IHÉS in 2000 in celebration of 6Nancago refers to the two cities where Weil taught at the time: Gromov’s Balzan Prize, was recorded by François Tisseyre and is Nancy and Chicago. available as a free video at https://vimeo.com/188585117.

1290 Notices of the AMS Volume 64, Number 11 Dennis Sullivan to some neglected important point. He strongly felt that lectures should be accessible to beginners, at least Marcel Berger and his wife, Odile, provided a warm family whenever possible. atmosphere for the scientists at IHÉS during the years of Of course, Marcel’s former students all eventually his directorship of the Institut des Hautes Études Scien- became aware of the game he was playing. Nonetheless, tifiques. I fondly remember Odile welcoming us to their when asking such questions he could manage to appear apartment overlooking the Champ-de-Mars while noticing so bewildered that we often caught ourselves being taken Marcel’s patient tolerance of my American manners. These in by the performance. He wasn’t just an outstanding personal contacts introduced me to many aspects of the mathematician—he was an amazingly good actor! French tradition—not only to such mathematical matters Now that he has left us, these feigned expressions as the history of differential geometry in France but also of bewilderment remain vividly alive in my memory and to some of the charming eccentricities of everyday life in somehow encapsulate the essential benevolence of his that country. attitude towards everyone around him. I have met mathe- Marcel was an able leader of the IHÉS. On his watch, maticians whose technical skills were more highly praised important matters were mainly decided by consensus, or highly prized, but I have never met a mathematician but I remember one occasion when he felt compelled who had so thoroughly earned the unalloyed admiration to override the deliberations of the Scientific Committee, and affection of his students. vetoing a proposal on very reasonable, common-sense Having now told you something about Marcel, I will grounds. conclude by telling you the story of his frequently Much later, in mathematical activities involving col- misunderstood collaborator leagues here at Stony Brook, I came to truly appreciate Arthur L. Besse (1975–?): the deep and beautiful contribution Marcel made to the Mathematical Knight of the Auvergne subject of special holonomy. These results, elegantly re- Marcel was literally obsessed by certain questions formulated in the thesis of Jim Simons, have led to a that he judged to be natural and important: Riemannian flourishing key area of current research in mathematics analogs of Kac’s celebrated problem on “hearing the and physics. They contribute to the unifying theme un- shape of a drum,” isosystolic inequalities, and classifying derlying activities at our Simons Center for Geometry and Riemannian manifolds whose geodesics are all closed and Physics and also to certain international collaborations of the same length. In the early 1970s, one seemed to funded by the Simons Foundation. know surprisingly little about this last question, despite the availability of some key evidence: Jacques Lafontaine • the series of examples given by the rank-1 symmetric spaces, Since many people • the existence of metrics on spheres that enjoy this have already written property but are geometrically different from the so eloquently about He strongly felt that obvious “round” example, and Marcel Berger’s math- • the fact that any manifold admitting such a metric ematical work, I will lectures should be must have the cohomology ring of a rank-1 symmetric not try to expand fur- accessible to space. ther on that theme, but will instead con- Marcel thus judged the time to be ripe for a systematic beginners. attack on the problem and therefore, in May 1975, orga- fine myself to a 8 few remarks about nized a workshop in the village of Besse-en-Chandesse, Marcel’s personality. located in the highlands of the Massif Central region of Marcel enjoyed attending mathematical lectures, and south-central France. The meeting took place at a hotel when he did so he always asked many questions. But named Les Mouflons, after the wild mountain sheep that these questions often focused on elementary points and wander the surrounding Volcans d’Auvergne Natural Park, could sometimes seem so naïve that someone might be under a sky full of towering cumulo-nimbus clouds. Both tempted to blurt out, “Marcel, how is it possible that you of these features of the area were later mentioned by the don’t know that?” pseudonymous Arthur L. Besse in the cryptic introduc- However, he eventually confessed to me that he often tion to his first mathematical publication. Even though asked questions whose answer he knew perfectly well, the locale was reasonably easy to reach from Paris or either just to slow down the lecturer or to draw attention Grenoble, this tranquil, bucolic setting seemed a world away from the participants’ daily routine, and we felt a Dennis Sullivan is Albert Einstein Professor of Mathematics certain quiet pride in being able to show the foreigners at the CUNY Grad Center and Distinguished Professor of in our midst around this picturesque village, with its Mathematics at Stony Brook University. His email address is remarkable natural and architectural environment. While [email protected]. some foreign participants initially seemed surprised to be Jacques Lafontaine is professor emeritus of mathematics at the University of Montpellier, France. His email address is 8Traditionally simply called Besse, the village is now part of a [email protected]. larger municipality called Besse-et-Saint-Anastaise.

December 2017 Notices of the AMS 1291 spending so much of their time at the table, the food was Manifolds [20], which differs markedly from its predeces- outstanding, and the delightful local cheeses seemed to sor in both style and scope; while the first book might be help put our discussions on a fast-track to mathematical called a monograph, the second could better be termed discovery. a treatise. Preparations for this new effort obliged Besse In view of the meeting’s overwhelming success, Mar- to leave his native Auvergne for the adjoining Rouergue cel soon urged the participants to cast their newfound region, where a meeting at Espalion, 100 miles south of understanding in permanent form by jointly authoring a Besse-en-Chandesse, resulted in a broad outline of the book. A consensus emerged that the group should write new book in September 1977. The intellectual pleasures of under a collective pseudonym, thereby following (in this our mathematical discussions were once again reinforced respect only) the earlier model provided by Bourbaki. The by those of the local gastronomy, and the weather, far last name of Besse was chosen in homage to the site of milder than that at our previous meeting, allowed for a our fruitful meeting. Because the CNRS had designated memorable excursion to the medieval pilgrimage site of the workshop as a “round table” for funding purposes, it Conques, some fifty kilometers to the west. was natural to invoke the legend of Camelot to provide The author’s royalties from the first book helped the first name of Arthur. To decrease the probability of underwrite the working meetings that made the second a lawsuit being brought by some real-life Arthur Besse, it one possible. Thicker, more classic, and less rococco was then decided that the name should be further embel- than its predecessor, this second book appeared in 1987, lished by the addition of an American-style middle initial, exactly at a moment when a conjunction of scientific abbreviating the undisclosed middle name of Lancelot. developments highlighted its many deep connections Such was the birth of Arthur L. Besse. to the mainstream of differential geometric research. It As soon as the meeting continues to be a key reference even now and has become ended, Arthur and his collab- Arthur Besse’s best known and most widely cited work. The writing orators got to work. The book, And what of a third book? Well, it was never in the had a entitled Manifolds All of Whose cards for a variety of reasons. In the 1970s Besse’s Geodesics Are Closed [19], was collaborators were largely concentrated at Jussieu and catalytic published by Springer in 1978. in the surrounding neighborhoods of Paris; for example, The writing had a catalytic ef- the École Polytechnique was still located in Paris, on effect. fect; as the book took shape, the nearby rue Descartes. But little by little, the group significant new results were dispersed to the suburbs and to other regions of France. obtained by members of the The international model of publish or perish also began group, and these were then added to the book as ap- to take hold in France, making it far more difficult to pendices under the names of the individual authors. In convince mathematicians (and especially young ones) one such case, the British author and his French-teacher to work quasi-anonymously, focusing on the beauty of wife submitted an appendix carefully written in beautiful mathematics rather than on their own personal glory. French, only to be met with implacable demands from the For all these reasons, Arthur Besse eventually be- publisher that it be translated into English! However, in came content to simply lend his name to a seminar in the end, we did manage to wrangle one major concession: differential geometry, first at Jussieu and later at the the book’s introduction was published in French and takes École Polytechnique’s new location in Palaiseau. However, the form of a cryptic letter bearing the signature of Arthur 9 Arthur continued to play a role in organizing conferences L. Besse, ostensibly writing from his estate at Le Faux. from time to time. These activities have left a durable Besse’s letter begins with some brief musings on a visit mark on mathematics in the form of various publications he received four decades earlier from a mathematically 10 bearing the Besse imprimatur [1], [15], [21], [22]. inclined general who had described both his grandiose One of Arthur Besse’s last public acts was to organize a vision of the whole of mathematics and his plans for conference at the CIRM in Luminy, honoring both the twen- a treatise on differential geometry that had somehow tieth anniversary of Einstein Manifolds and the eightieth never come to fruition. The practical joke then continues birthday of Besse’s true father, Marcel Berger. Speaking with a wry mixture of fact and fiction concerning French on behalf of Springer-Verlag, Arthur’s old friend Catriona mathematics, Besse’s beloved home, and the contents of Byrne made the surprise announcement that a second the book. edition of Einstein Manifolds had been published and then But Arthur Besse didn’t stop there. Pleased with the proceeded to provide each conference participant with a success accorded to his first foray into collaborative writ- free copy. But while the conference center, located amidst ing, he then went on to write a second book, Einstein the limestone cliffs that line the coast south and east of Marseille, was both functional and pleasant, it also served 9The name of this real place, located roughly a kilometer south- west of Besse-en-Chandesse, means The False or The Forgery in as a sign of the times, reminding some of us that those French. heady days of intimate meetings at gastronomical hotels 10General Nicolas Bourbaki and Arthur L. Besse share an im- had, alas, now become a thing of the past. portant commonality that is rarely noted. Bourbaki’s founding As for Arthur, it seems that he has now definitively conference was held July 10–17, 1935, in Besse-en-Chandesse. retired to a life of leisure. But his beneficent presence Yes, really! will long remain with us, because the royalties generated

1292 Notices of the AMS Volume 64, Number 11 by his books have been donated in perpetuity to the I learned of this from a conversation with Jean-Pierre European Mathematical Society for the sole purpose of Bourguignon at a 1977 geometry conference in Berlin. I providing financial support to young mathematicians in was later pleased to be able to prove [18], [19] the desired developing countries. inequality. My contribution primarily relies on convexity inequal- Jerry Kazdan ities. It would be useful to find a more conceptual (and less technical) proof. Marcel Berger was a world leader in differential geometry. He had Marie-Louise Michelsohn inspiring geometric vision, ask- He had ing fertile questions that give inspiring Over the years, I have always had immense fondness and insight to techniques involving respect for Marcel Berger. I was truly saddened to learn both geometry and analysis. geometric of his death, and it is not an exaggeration to say that I In addition to his own funda- miss him. mental research, his seminars vision. I would like to communicate a side of him that many and projects attracted a remark- may not know by an anecdote that has stayed with me able group of younger mathematicians, primarily in France these many years and that I think of frequently. but also worldwide. Some of the fruits are collected in About three decades ago Marcel and I were correspond- books, both his own and by his group, “Arthur Besse.” ing mathematically, which at that time was done by postal He welcomed younger mathematicians. Of one early service and hand-written letters. In his first letters, he trip I made to Paris, I warmly remember a lunch in his addressed me as vous, as did essentially all French male apartment. He and his wife, Odile, were splendid hosts, mathematicians when addressing their female colleagues. warm and interesting. In a later letter, I finally requested that he address me as In both 1992 and 1994 he visited our department at tu, but he then replied that this would be “contre mon the University of Pennsylvania. His lectures and conversa- éducation.” I replied that to use vous for women was tions were widely appreciated by everyone, including our to exclude them from the confrérie and so was not at graduate students. all positive. It should be remembered here that Marcel’s My main direct mathematical interaction with Marcel upbringing was particularly conservative. Nonetheless, he concerned manifolds. Intuitively, these are understood right away, and until his death I was therefore 푛 compact Riemannian manifolds (푀 , 푔) without boundary always tu. with the property that at any point (think of the north This is a small illustration of the extraordinary char- pole of the standard round sphere) if a number of people acter of Marcel Berger. I am greatly saddened by his beginning in different directions walk along geodesics at death. the same constant speed, then after a certain amount of time they all meet again (think of the south pole). The Pierre Pansu conjecture was that this is only possible on the standard round sphere. It had been proved in dimension 2. One day in October 1979, at Misha Gromov’s first lecture Berger had the vision that a key ingredient in the proof at Paris VII, Marcel Berger asked the audience if anyone of the higher-dimensional cases would involve a sharp would volunteer to take notes on the lecture series, with isoperimetric inequality. At a point 푝 of 푀 consider a the idea that these notes might eventually form the small disk of radius 푟. As 푟 gets larger, the disk eventually basis for a book. I raised my hand. In many ways, this begins to overlap itself (think of a small disk on the simple, spontaneous act determined much of my future: I surface of a two-dimensional torus in ℝ3 and gradually would become a geometer, I would benefit from Gromov’s increase the radius of this disk). This largest radius is influence, I would participate in collective book projects, called the injectivity radius at 푝. The largest radius 휌 that and I would receive a great deal of personal attention works for all points 푝 on (푀푛, 푔) is called the injectivity from Marcel Berger. radius of (푀푛, 푔). For the standard round sphere of radius Berger’s fascination with Gromov’s achievements 푐 in ℝ푛+1 it is the geodesic distance between antipodal meshed perfectly with his more general enthusiasm for points: 휌 = 휋푐. geometry. Nevertheless, he was also fond of saying that Berger conjectured: every deep result in geometry should entail some serious Isoperimetric Inequality. If (푀푛, 푔) has injectivity ra- input from analysis. This precept has sometimes served dius 휌, then its volume is at least as large as the volume of as a counterweight to the style of thinking that I learned the round sphere with the same injectivity radius. Equality from Gromov. holds if and only if the manifold is isometric to this round Berger considered collective authorship to be a perfect sphere. training ground for young mathematicians. Fortunately, By a beautiful geometric argument, Berger reduced the proof to a quite specific sharp analytic inequality. Marie-Louise Michelsohn is professor of mathematics at Stony Brook University. Her email address is [email protected]. Jerry Kazdan is professor of mathematics at the University of Pierre Pansu is professor of mathematics at Paris-Sud University, Pennsylvania. His email address is [email protected]. France. His email address is [email protected].

December 2017 Notices of the AMS 1293 he was a model co-author. Although hardly a model pupil perhaps because he had largely stopped doing his own as a child—he was allegedly expelled from 11 schools!— research by that time. he was the only one in our “class” of twenty co-authors To me, Berger was a kind, soft-spoken man of fine, to consistently turn in his “homework” (book chapters) strong good will. I will miss him, and I think we all will. on time. His chapters sometimes contained key original results, and this enhanced the value of our work, because Karsten Grove he would never have bothered to publish these results under his own name. I feel enormously privileged and fortunate that, at an Berger exercised considerable care in checking (and early stage of my career, two of my idols, M. Berger and improving) various proofs in my thesis. Although he W. Klingenberg, both chose to take me under their wing. I claimed to simply be doing his duty, this was an incredible first met Berger at a small workshop in Bonn in the fall of favor. During the decades that followed, his attention from 1971, where Berger, Klingenberg, their students, and their afar continued to provide me with a key source of cheer visitors accounted for the vast majority of the participants. and encouragement. Fortunately, I later had the pleasure of seeing Berger many I never regretted raising my hand. more times, not only at Oberwolfach but also in Paris, at Luminy, and, most memorably, at the remarkable Table David Ebin Ronde workshop held at Besse-en-Chandesse in 1975. The topic of the Besse conference was Riemannian Berger and I first met in 1968 at an AMS Summer Institute manifolds with periodic geodesic flow, and stellar pre- in Berkeley. I had heard of him and was interested in sentations by Berger and his school concerning such meeting him. I saw someone I thought might be Berger manifolds provided the main focus of the activity there. chatting with other mathematicians, and so I asked, “Are This was the germ of what later evolved into Arthur you Berger?” He replied, “Yes, but there are two.” Since I Besse’s first book, Manifolds All of Whose Geodesics Are knew only his first initial, I then tried to be more precise, Closed [19], which to this day covers much of what is asking, “M. Berger?” And again he said,“Yes, but there are known on the topic. This was later followed by two two.” Eventually, I found out that the other one was Mel more Besse books, Einstein Manifolds [20] and Géométrie Berger,11 who was also at the Summer Institute. riemannienne en dimension 4 [1]. Berger was interested in variations of structures on Among the many profound and diverse contributions Riemannian manifolds due to a change in the Riemannian Berger made to Riemannian geometry, those that most metric. I had recently written my thesis on the space of inspired and affected my own work were the ones Riemannian metrics on a manifold and on the natural concerning manifolds with positive sectional curvature. action of the diffeomorphism group of the manifold Prior to 1960, the only known (simply connected) man- on this space. The thesis gave a convenient way to ifolds of positive curvature were the rank-one symmetric distinguish between changes in the metric that were due spaces, i.e., the standard spheres and projective spaces. to diffeomorphisms and those that represented actual Moreover, the celebrated Rauch–Berger–Klingenberg 1/4- changes in the geometry. This turned out to be what Pinching Theorem shows that, up to homeomorphism,12 Berger was looking for, so we discussed the matter and these are the only such manifolds with curvature varying soon produced a joint paper [16]. between 1 and 4; and if the manifold is not topologically Three years later, Berger and ar- a sphere, Berger moreover proved that it must actually ranged for me to give a semester of lectures at Paris be isometric to a rank-one symmetric space. The fact VII and the École Polytechnique. They also found lodging that these spaces all have diameter at least 휋/2 helped for my wife and me in the IHÉS Residence, and it was inspire my own proof, with Shiohama and Gromoll, of most enjoyable. My wife, who is an amateur violinist, the so-called Diameter Sphere Theorem. In conjunction played in a string quartet at the Château de l’Ermitage, with a 2001 result of Wilking, this says that a complete in nearby Gif-sur-Yvette. The Bergers also invited us to Riemannian manifold with sectional curvature ≥ 1 and dinner twice—first with their family, and later to what diameter ≥ 휋/2 is finitely covered by either a topological might today be called a mathematician’s power dinner, sphere or a compact rank-one symmetric space. with the Cartans and the Thoms. In the early 1960s, Berger found two additional exam- I next saw Berger at another AMS Summer Institute, in ples [5] of positively curved manifolds by studying a class Stanford. There he gave a series of lectures on the work of spaces called normal homogeneous manifolds. Since of Colin de Verdière. It used the heat equation to get then, many other manifolds of positive sectional curva- information about geodesics and frequently involved the ture have been found, with some new construction being fundamental solution of the heat equation, FSHE, which discovered every decade or so. A common feature of all Berger referred to as fish. After that, Berger came to visit known constructions is that they produce examples with our department in Stony Brook several times. During his last visit, he gave a course on history of mathematics, Karsten Grove is Kenna Professor of Mathematics at Notre Dame University. His email address is [email protected]. David Ebin is professor of mathematics at Stony Brook University. 12A 2009 paper by Brendle and Schoen shows that the conclusion His email address is [email protected]. even holds up to diffeomorphism; thus, an exotic sphere can never 11Melvyn Berger’s surname is pronounced Burger. admit a 1/4-pinched metric.

1294 Notices of the AMS Volume 64, Number 11 relatively large groups of symmetries, and a great deal [11] , Une borne inférieure pour le volume d’une variété of subsequent work has therefore focused on classifying riemannienne en fonction du rayon d’injectivité, Ann. Inst. positively curved manifolds with large isometry groups. Fourier (Grenoble) 30 (1980), 259–265. MR597027 One of the basic tools used in this area is also due to [12] , Sur les variétés riemanniennes pincées juste au- dessous de 1/4, Ann. Inst. Fourier (Grenoble) 33 (1983), Berger, who proved [8] that any Killing field (vector field 135–150 (loose errata). MR699491 generating a one-parameter group of isometries) on an [13] , Riemannian geometry during the second half of even-dimensional positively curved manifold must have the twentieth century, Jahresber. Deutsch. Math.-Verein. 100 a zero; similarly, in odd dimensions, two Killing fields (1998), 45–208. MR1637246 will be dependent at some point. Yet another important [14] , A Panoramic View of Riemannian Geometry, tool in this context, often called the Cheeger deformation, Springer-Verlag, Berlin, 2003. MR2002701 has its roots in a deformation introduced by Berger in his [15] , Yves et Arthur: quelques souvenirs, Festival Yves Colin de Verdière, Ann. Inst. Fourier (Grenoble) 57 (2007), analysis of what are now called Berger spheres. 2083–2089. MR2394536 As a supportive colleague with a positive and friendly [16] M. Berger and D. Ebin, Some decompositions of the outlook, Berger will be sorely missed. But as an exceptional space of symmetric tensors on a Riemannian manifold, mathematician with a tremendous impact, he will never J. Differential Geometry 3 (1969), 379–392. MR0266084 be forgotten. [17] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971. MR0282313 [18] M. Berger and J. L. Kazdan, A Sturm-Liouville inequality EDITOR’S NOTE. See also Berger’s article on with applications to an isoperimetric inequality for volume Gromov in the February 2000 Notices, “Encounter in terms of injectivity radius, and to wiedersehen mani- with a Geometer,” www.ams.org/notices/200002 folds, in General Inequalities, 2 (Proc. Second Internat. Conf., /fea-berger.pdf and “WHAT IS...a Systole?” in Oberwolfach, 1978), Birkhäuser, Basel-Boston, Mass., 1980, the March 2008 Notices, www.ams.org/notices pp. 367–377. MR608261 /200803tx080300374p.pdf. [19] A. L. Besse, Manifolds All of Whose Geodesics Are Closed, with appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard- Bergery, M. Berger, and J. L. Kazdan, vol. 93 of Ergebnisse der Berger’s major surveys [13], [14] are highly recom- Mathematik und ihrer Grenzgebiete [Results in Mathematics mended. and Related Areas], Springer-Verlag, Berlin-New York, 1978. MR496885 Cited Works by Berger and His Collaborators [20] , Einstein Manifolds, vol. 10 of Ergebnisse der Mathe- [1] L. Bérard-Bergery, M. Berger, and C. Houzel, eds., matik und ihrer Grenzgebiete (3) [Results in Mathematics and Géométrie riemannienne en dimension 4, Papers from the Related Areas (3)], Springer-Verlag, Berlin, 1987. MR867684 Arthur Besse seminar held at the Université de Paris VII, Paris, [21] , Some trends in Riemannian geometry, in Duration 1978/1979, vol. 3 of Textes Mathématiques [Mathematical and Change, Springer, Berlin, 1994, pp. 71–105. MR1322983 Texts], CEDIC, Paris, 1981. MR769127 [22] , ed., Actes de la Table Ronde de Géométrie Dif- [2] M. Berger, Sur les groupes d’holonomie des variétés rie- férentielle, En l’honneur de Marcel Berger. [In honor of manniennes, C. R. Acad. Sci. Paris 237 (1953), 472–473. Marcel Berger], Held in Luminy, July 12–18, 1992, vol. 1 MR0056346 of Séminaires et Congrès [Seminars and Congresses], So- [3] , Sur les groupes d’holonomie des variétés riemanni- ciété Mathématique de France, Paris; distributed by American Mathematical Society, Providence, RI, 1996. MR1427752 ennes non symétriques, C. R. Acad. Sci. Paris 237 (1953), 1306–1308. MR0059046 [4] , Les variétés Riemanniennes (1/4)-pincées, Ann. Photo Credits Scuola Norm. Sup. Pisa 14 (1960), no. 3, 161–170. Photo of Marcel Berger and photo in Besson section courtesy of MR0140054 JFD/IHÉS. [5] , Les variétés riemanniennes homogènes normales Photo of Marcel Berger in the Cheeger section ©Mathematisches simplement connexes à courbure strictement positive, Forschungsinstitut Oberwolfach Photo Collection; photo by Ann. Scuola Norm. Sup. Pisa 15 (1961), no. 3, 179–246. Konrad Jacobs, Erlangen. MR0133083 Photo of Marcel Berger in the Bourguignon section ©Mathema- [6] , Sur les variétés à courbure positive de diamètre min- tisches Forschungsinstitut Oberwolfach Photo Collection; imum, Comment. Math. Helv. 35 (1961), 28–34. MR139121 photo by Dirk Ferus, Berlin. [7] , An extension of Rauch’s metric comparison theorem and some applications, Illinois J. Math. 6 (1962), 700–712. MR0143159 [8] , Trois remarques sur les variétés riemanniennes à courbure positive, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A76–A78. MR0199823 [9] , Sur certaines variétés à géodésiques toutes fermées, Bol. Soc. Brasil. Mat. 9 (1978), 89–96. MR552030 [10] , Rapport sur les variétés d’Einstein, in Analysis on Manifolds (Conf., Univ. Metz, Metz, 1979) (French), vol. 80 of Astérisque, Soc. Math. France, Paris, 1980, pp. 3, 5–19. MR620166

December 2017 Notices of the AMS 1295 A MERICAN M ATHEMATICAL S OCIETY

THE 2018 AMS EMPLOYMENT CENTER at the Joint Mathematics Meetings Hall A San Diego Convention Center, San Diego California January 10–13, 2018

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Open hours: Wednesday, January 10, 8:00 am – 5:30 pm Thursday, January 11, 8:00 am – 5:30 pm Only interviewing Friday, January 12, 8:00 am – 5:30 pm Saturday, January 13, 9:00 am – 12:00 pm a few people? Visit www.ams.org/emp-reg for registration instructions. Try a ONE DAY TABLE THE GRADUATE STUDENT SECTION

Ciprian Manolescu Interview Conducted by Alexander Diaz-Lopez

Diaz-Lopez: When did you know you wanted to be a math- ematician? Manolescu: As a kid I liked math puzzles and games. I remember being particularly fond of a book by Martin Gardner (in Romanian translation). In middle school, I started participating in math olympiads, and by then I figured I wanted to do mathematics in some form when I grew up. At the time I did not know any research math- ematicians personally, but I read a couple of books popu- larizing mathematical research and news articles about Wiles’ proof of Fermat’s Last Theorem. It all sounded very exciting. Diaz-Lopez: Who encouraged or inspired you? Manolescu: My parents have been extremely support- ive, and I was fortunate to have good teachers—especially my high school math teacher Stefan Alexe, who lent me a large part of his collection of problem books to prepare for the olympiads. Clearly, however, my greatest math- ematical debt is to my college and PhD advisor, Peter Kronheimer. We had weekly meetings for several years, and I learned a great deal of mathematics from him. Diaz-Lopez: How would you describe your work to a graduate student? Manolescu: I work in topology, where the main motivat- ing problem is the classification of smooth shapes (mani- folds). There is a theorem that a complete classification is not possible in dimensions four or higher, because we run into group-theoretic difficulties. However, one can focus on manifolds with a reasonable fundamental group, and Ciprian Manolescu is professor of mathematics at then the problem becomes tractable in dimensions five the University of California, Los Angeles, working in or higher. Dimensions three and four are the interesting topology and geometry. He is a winner of the Morgan ones. In dimension four, most questions are still open, but Prize and the European Mathematical Society Prize, a Fellow of the American Mathematical Society, and an (starting with the work of in the 1980s) invited speaker at the 2018 International Congress there has been steady progress on this topic, using gauge of Mathematicians. He is the only person to achieve theory. Mathematical is the study of a certain three perfect scores at the International Mathematical kind of partial differential equations (Yang–Mills, Seiberg- Olympiad. Witten) that originated in theoretical physics. My research focus is on Floer homology, which is a way of capturing the gauge-theoretic information for a three-dimensional mani- For permission to reprint this article, please contact: fold. The ultimate goal is to understand four-dimensional [email protected]. manifolds, and Floer homology tells us something about DOI: http://dx.doi.org/10.1090/noti1602 what happens when we cut the four-dimensional mani-

DeceMber 2017 Notices of the AMs 1297 THE GRADUATE STUDENT SECTION

Figure 1: The unknot detection algorithm based on knot Floer homology starts by representing the knot by a grid diagram: an n-by-n grid with one O marking and one X marking in each row and in each column, such that if we join the Os to the Xs by vertical and horizontal arcs (with the vertical arcs on the top), we obtain the knot. The figure above shows how to transform a planar projection of the figure-eight knot into a grid diagram. fold along a three-dimensional one. Floer homology also grad school, so that you have time to write a good thesis has applications to three dimensions per se, and to knot (and maybe a few other papers). Start attending the re- theory. For example, with my collaborators Peter Ozsváth search seminars in your area as soon as you can—even if and Sucharit Sarkar, we used Floer homology to describe a you don’t understand the talks, just getting an idea of what new algorithm (see Figure 1) for detecting when a tangled- people are working on is helpful. At the same time, after up piece of string can be unknotted (without breaking it). you decide on your field, do keep learning mathematics Diaz-Lopez: What theorem are you most proud of and in other areas. This will help you in the future, as many what was the most important idea that led to this break- fruitful projects come from making connections between through? different parts of mathematics. With regard to writing Manolescu: The disproof of the high-dimensional your thesis, keep in mind that it is normal for it to take a triangulation conjecture. The question of whether all long time; the transition from solving homework problems manifolds can be triangulated was raised by Kneser in to doing research is difficult for everyone. 1924. The work of (based on that of Mike Diaz-Lopez: All mathematicians feel discouraged oc- Freedman) in the 1980s provided counterexamples in casionally. How do you deal with discouragement? dimension four. In dimensions five or higher, the ques- Manolescu: There are a few types of disappointments tion had been reduced (by David Galewski, Ronald Stern, that appear regularly in a mathematician’s life. The first is and Takao Matumoto) to a different problem, about the when you cannot solve the problem you’re working on. It three-dimensional homology cobordism group. Thus, one is then good to remember that this is expected—perhaps could hope to use the techniques of gauge theory to solve the problem just cannot be solved with current techniques; it. I knew about the problem for a while, but I only started the miracle is when you solve it! The second type is when thinking about it seriously in 2012, and it came to my at- you think you solved a problem, you start writing it up, tention in a roundabout way. I had an idea about how to and then you run into a technical difficulty, which might construct a version of an invariant of Casson’s, and I was invalidate the whole thing. This happens to me quite often, asking other mathematicians about what kind of applica- and it is disconcerting to think that all the previous work tions one could hope for from such a construction. Sev- I did on a project could be in vain. However, so far I've eral mentioned the triangulation problem, as something almost always managed to solve this kind of issue, after that Casson’s invariant helped solve in dimension four some more work. By now I know that if something looks but not in higher dimensions. It soon became clear that like a technical but not a fundamental difficulty, it prob- the version I had in mind would be of no help with this ably is—so I should just be pushing harder to overcome it. question. Nevertheless, I then became curious about what Diaz-Lopez: You have won several honors and awards. kind of invariant one needs in order to solve the problem. Which one has been the most meaningful and why? I remembered a different invariant from the literature, due Manolescu: The Prize of the European Mathematical to Kim Frøyshov, which came closer to what was needed. Society, which I received in 2012. It came as a nice surprise, After some thought, I realized that if one incorporated at a time when I was getting bogged down in some long and into the picture a further symmetry of the Seiberg-Witten technical research projects. The prize gave me the courage equations, based on the non-abelian group Pin(2), then and confidence to think about some other fun problems one could solve the triangulation problem. on the side, including the triangulation conjecture—which Diaz-Lopez: What advice do you have for graduate I disproved eight months later. students? Diaz-Lopez: Is being the only person to achieve three Manolescu: Choose a good advisor, by talking to the perfect scores at the International Math Olympiad a close older students and asking them about their experiences; second? also, find out what kind of jobs the past students of each Manolescu: This was certainly important for me in high potential advisor got after graduation. Pick a subfield of school, and it helped with my career path, since it enabled mathematics that you enjoy and try to specialize early in me to get a scholarship to study at Harvard. I should say

1298 Notices of the AMS Volume 64, Number 11 that my performance involved a good amount of luck. I prepared a lot for the contests, but it is not the case that I could have solved any Olympiad problem. For example, I did not solve all the problems on the Romanian selection tests for the IMO. It just happened that I managed to see the solutions to the IMO problems, the three times that I participated. Diaz-Lopez: If you could recommend one book to gradu- ate students, what would it be? Manolescu: Milnor’s Morse Theory is the book that turned me towards geometry and topology. It is short, clear, and beautifully written. It not only covers Morse theory (which lies at the basis of the study of manifolds), but also touches upon several other useful topics, from the Lefschetz theorem in algebraic geometry to Bott peri- odicity. Chapter 2 is the place from where I first learned the basics of .

Image Credits Photo of Ciprian Manolescu courtesy of Reed Hutchinson. Figure 1 based on Figure 7 of “An introduction to knot Floer homology,” Physics and Mathematics of Link Homology 680 (2016), Amer. Math. Soc. and CRM.

ABOUT THE INTERVIEWER Alexander Diaz-Lopez, having earned his PhD at the University of Notre Dame, is now assistant professor at Villanova University. Diaz-Lopez was the first graduate student member of the Notices Edi- torial Board. Alexander Diaz-Lopez

Graduate students at member institutions can opt to receive print copies of the Notices by updating their individual member profiles at www.ams.org/cml/update-ams

December 2017 Notices of the AMS 1299 THEGRADUATESTUDENTSECTION

WHATIS… a Scheme? Tom Gannon Communicated by Cesar E. Silva

Schemes pop up everywhere in mathematics. But getting But what exactly is an affine scheme? Affine schemes a grasp on this ubiquitous concept is not so easy. And the are the full manifestation of a simple idea: If 퐴 is a bare definition might leave you wondering why a scheme commutative ring, we should view the elements of that is defined the way it is. The fol- ring as functions instead of points. If you’re thinking lowing interpretation introduces about rings like ℚ(푖) or ℤ/6ℤ, this can be a little weird, The idea of important elements in the defini- but if you do the construction by first imagining rings tion of a scheme and helped me like ℝ[푥, 푦] it might be more clear. But the beautiful part a scheme understand the basic idea. I hope it is, this idea works for any ring, meaning that any ring can borrows can help you too. be viewed as functions on some space. Schemes have played a funda- For the remainder of this article, fix a commutative from the mental role in algebraic geometry ring 퐴. You might ask, “What space could this ring 퐴 ever since they were introduced by be a function on?” To motivate this, let’s do an example idea of a [Gr]. The from basic high school algebra. If you want to see where 3 2 idea of a scheme borrows from the the polynomial 푝(푥) ∶= 푥 + 푥 + 푥 vanishes, your natural 2 manifold. idea of a manifold in differential instinct would be to factor: 푝(푥) = 푥(푥 + 푥 + 1). Then if 2 topology. Recall that a manifold is 푝(푥) = 0, you know that either 푥 = 0 or 푥 +푥+1 = 0. Put in a topological space locally diffeomorphic to an open slightly more abstract terms, if the function 푝(푥) vanishes 푛 at some point, then either the function 푥 vanishes at that subset of ℝ , subject to some additional topological 2 considerations to avoid pathologies. Now, if you have a point or the function (푥 + 푥 + 1) vanishes at that point. This is an important property we’d like our topological function defined on an open neighborhood of a point in space to satisfy: ℝ푛, you can take the derivative of that function at that The “Vanishing” Property: If 푓, 푔 ∈ 퐴 and the product point. So a manifold is a topological space that locally 푓푔 vanishes at a point 푝 in the topological space, then looks like “the place to do differential calculus.” In a simi- either 푓 vanishes at 푝 or 푔 vanishes at 푝. lar way, a scheme is a topological space that locally looks This definition looks suspiciously close to the definition like “the place to do algebra/find zeros of functions.” But of a prime ideal of a ring. Recall that a prime ideal 픭 of that’s a pretty long name, so let’s call “the place to do a ring 퐴 is an ideal where if 푓, 푔 ∈ 퐴 and 푓푔 ∈ 픭, then algebra/find zeros of functions” an affine scheme instead. either 푓 ∈ 픭 or 푔 ∈ 픭. So if we assume that the “points” (Note for advanced readers: There’s actually more to it of our special topological space are prime ideals of the than just a topological space. There is a sheaf of functions ring 퐴, then we have an obvious choice for what it might associated to the space as well, but we won’t go into that mean for a function 푓 ∈ 퐴 to vanish at the point 픭. We’ll here.) say 푓 vanishes at 픭 if 푓 ∈ 픭 (or equivalently, it is zero in the ring 퐴/픭). Tom Gannon is a graduate student in algebraic geometry at the Given a ring 퐴, define the spectrum of 퐴, written University of Texas at Austin. His email address is gannonth@math. 푆푝푒푐(퐴) to be the set of prime ideals of 퐴. (Again, for utexas.edu. advanced readers: We’ve just defined a scheme in the For permission to reprint this article, please contact: category of sets, but you can put additional structure [email protected]. on your spectrum so that it becomes an object in the DOI: http://dx.doi.org/10.1090/noti1603 category of locally ringed spaces.) Now I’ve promised a

1300 Notices of the AMS Volume 64, Number 11 THEGRADUATESTUDENTSECTION topology on this space, and any good topology in a This is about half of the definition of an affine scheme. subject relating to “vanishing” should have the topology Continuing with the “rings are functions on some natural related to vanishing. And luckily for us, it does. We space” interpretation, we can develop a sheaf on the define a set of points (which are prime ideals, but we’re space 푆푝푒푐(퐴) so that we can, for example, about the thinking of them as “points”) to be closed if it is of the 푥2+푦2 function 푥−3 on the places where 푥 − 3 doesn’t vanish. form 푉(푆) = {픮 ∈ 푆푝푒푐(퐴) ∶ 푆 ⊂ 픮}, for some set of Sheaves are beyond the scope of this short piece, which “functions” 푆 ⊂ 퐴 (meaning, “every element of 푆 vanishes is intended only to give you a good picture of what the at 픮.”) Checking this forms a topology on 푆푝푒푐(퐴) is not topological space of an affine scheme looks like. And I too hard and is a good exercise. hope it has also given you a new interpretation of rings! This is a good definition for affine schemes (which, recall, Any ring can Further Reading was meant to be related to [Gr] A. Grothendieck (with J. Dieudonne), Eléments de determining where functions be viewed as Géométrie Algébrique, Publ. Math. IHES, Part I:4 (1960); Part vanish) because in some sense II:8 (1961); Part III:11 (1961), 17 (1963); Part IV:20 (1964), 24 the definition is “maximal.” functions on (1965), 28 (1966), 32 (1967). MR 0217083 That is, any set of elements [Ha] R. Hartshorne, Algebraic Geometry, Grad. Texts in in the ring 퐴 that can be the some space. Math. 52, Springer-Verlag, New York. Heidelberg, 1977. set of functions vanishing at a MR 0463157 point is a prime ideal (since it has the “Vanishing” Property). Moreover, functions dis- EDITOR’S NOTE. This article is a revision of Gannon’s tinguish the points in the topological space. That is, if post “The ‘Idea’ of a Scheme” on the AMS Graduate Stu- two points 픭, 픮 in the space are distinct, then there is a dent Blog blogs.ams.org/mathgradblog. A related function that vanishes on one of the points, but not on article “WHAT IS...an Anabelian Scheme?” by Kirsten the other. Seeing why this is true is a good check on your Wickelgren appeared in the March 2016 issue of understanding of the concept. However, given two points the Notices, www.ams.org/notices/201603/rnoti- 픭, 픮 ∈ 푆푝푒푐(퐴), it isn’t necessarily the case that you can p285z.pdf. find a function 푓 ∈ 퐴 vanishing at 픭 but not at 픮. This is a good exercise to check your understanding of the material, and an example is also provided below. Example: Consider the ring 퐴 ∶= ℂ[푥, 푦]. What are ABOUT THE AUTHOR some points in 푆푝푒푐(퐴)? Certainly if you choose some 푎, 푏 ∈ ℂ, then 픭 ∶= (푥 − 푎, 푦 − 푏) is a maximal ideal, When not doing mathematics, Tom since the quotient field 퐴/픭 is ℂ, a field, and thus it Gannon can often be found in is a prime ideal. A good way to think about points of the kitchen cooking new recipes, the form (푥 − 푎, 푦 − 푏) is to think of the points of the or can be found in the kitchen form (푎, 푏) ∈ ℂ2 (equivalently, the places on the 푥푦 cooking old recipes. plane where 푥 − 푎 = 0 and 푦 − 푏 = 0). The function 푝(푥, 푦) ∶= 푥4 + 푥3푦 − 푥푦 − 푦2 ∈ 퐴 vanishes at 픭 since Tom Gannon 푝(푥) = 푟(푥, 푦)(푥 − 2) + (−푥 − 푦)(푦 − 8) ∈ 픭 Photo Credit where Photo of Tom Gannon courtesy of Tom Gannon. 푟(푥, 푦) = 푥3 + 2푥2 + 4푥 + 푥2푦 + 2푥푦 + 4푦. Also, 푝(푥, 푦) vanishes in our normal sense because 푝(2, 8) = 24 + 26 − 24 − 26 = 0. So we can conclude that vanishing in our new sense generalizes vanishing in the old sense. Now the question becomes, what else can we get from this scheme construction? One quick payoff from the definition of affine schemes is that we can talk about functions vanishing at places that aren’t points in the traditional sense. For example, consider the polynomial 푞(푥, 푦) ∶= 푥3 − 푦. It is an irreducible polynomial by Eisenstein’s Criterion, so the ideal 픮 ∶= (푥3 −푦) is another point in 푆푝푒푐(퐴). This has an obvious interpetation in ℂ2. It’s the curve 푦 = 푥3. And more amazingly, notice that 푝(푥, 푦) ∶= 푥4 + 푥3푦 − 푥푦 − 푦2 = (푥3 − 푦)(푥 + 푦), so 푝(푥, 푦) ∈ 픮. We can interpret it to mean the function 푝(푥, 푦) vanishes on the entire curve. That is very pretty.

December 2017 Notices of the AMS 1301 THE GRADUATE STUDENT SECTION

The AMS Graduate Student Blog, by and for graduate students, includes puzzles and a variety of interesting columns, such as this one from August 2017: blogs.ams.org/mathgradblog.

Donaldson Turns 60 by Jacob Gross This month [August] marks [Simon Donaldson’s] 60th birthday. A conference1 on symplectic geometry will be held at the Isaac Newton Institute, Cambridge, to cel- ebrate this occasion and various advances in symplectic geometry. Below [in this blog post] we mention some major contributions of Donaldson to the ancient science of geometry; in particular, to gauge theory and to Kähler– Einstein metrics. ...

Photo Credits Photo of Simon Donaldson by Bob Giglione/Simon Center for Geometry and Physics. Photo of Jacob Gross courtesy of Jacob Gross.

ABOUT THE AUTHOR Jacob Gross is a graduate student in geometry at Oxford University, working under the supervision of Dominic Joyce FRS, and supported by the Simons Collaboration on Spe- cial Holonomy in Geometry, Analy- sis, and Physics. His email address is Simon Donaldson [email protected] . Jacob Gross

1 www.newton.ac.uk/event/sygw05

1302 Notices of the AMS Volume 64, Number 11 A m e r i c a n M a t h e m a t i c a l S o c i e t y

The American Mathematical Society welcomes applicants for the 2018 Mathematics Research Communities

TOPICS FOR 2018 June 3–9, 2018 Week 1a: The Mathematics of Gravity and Light Organizers: Charles Keeton (Rutgers University) Arlie Petters (Duke University) Marcus Werner (Kyoto University) The Mathematics Research Communities program helps early-career Week 1b: Harmonic Analysis: New Developments mathematicians develop long-lasting cohorts for collaborative on Oscillatory Integrals research projects in many areas of mathematics. Organizers: Philip T. Gressman (University of Pennsylvania) Larry Guth (Massachusetts Institute of Technology) - Intensive one-week, hands-on research conference in Lillian B. Pierce (Duke University) the summer (see the 2018 research topics to the right) June 10–16, 2018 - Special Sessions at the Joint Mathematics Meetings Week 2a: Quantum Symmetries: Subfactors and - Guidance in career building Fusion Categories - Follow-up small-group collaborations Organizers: David Penneys (The Ohio State University) - Longer-term opportunities for collaboration Julia Plavnik (Texas A&M University) Over time, the participants are expected to provide feedback about Noah Snyder (Indiana University) career development and the impact of the MRC program. Week 2b: Number Theoretic Methods in Hyperbolic Geometry The summer conferences of the Mathematics Research Organizers: Benjamin Linowitz (Oberlin College) Communities will be held at the Whispering Pines Conference David Ben McReynolds (Purdue University) Center in West Greenwich, Rhode Island, where participants can Matthew Stover (Temple University) enjoy the natural beauty and a collegial atmosphere. Women and underrepresented minorities are especially encouraged to apply. June 17–23, 2018 The application deadline for summer 2018 is February 15, 2018. Week 3: Agent-based Modeling in Biological and This program is supported by a grant from the National Science Social Systems Foundation. Organizers: Andrew Bernoff (Harvey Mudd College) Leah Edelstein-Keshet (University of British Columbia) Learn more at: www.ams.org/mrc Alan Lindsay (University of Notre Dame) Chad Topaz (Williams College) Alexandria Volkening (Mathematical Biosciences Institute at Ohio State) Lori Ziegelmeier (Macalester College) COMMUNICATION

Reducing Bias in Faculty Searches

Elizabeth A. Burroughs

Communicated by Stephen Kennedy

Historically, women and individuals from marginalized What Montana State Does populations have been underserved by the discipline of There are two foci in Montana State University’s plan1 mathematics, with the result that the knowledge and tal- to increase diversity in faculty hires: ents of these individuals have not been available to enrich (1) recognizing that everyone holds unintentional biases the discipline. Across the mathematical community, there that can influence our evaluation of applicants; and is a growing awareness about the need and opportunity to (2) purposefully focusing on recruitment to create a foster diversity. Individuals with diverse backgrounds can pool of applicants who are well-qualified and from diverse provide a department with new research perspectives or backgrounds. methods that enhance the research work of faculty with The university focuses on search committee training as similar interests, and may attract more diverse students a tool to give practical strategies to address both aspects and collaborators. These contributions to diversity can of this plan. Other programming across campus, such as fulfill underlying goals of a department in reaching more guest speakers and campus community discussions, have students and generating new mathematical knowledge. provided a forum to discuss the underlying reasons that having diversity on campus can be valuable. What Is Implicit Bias? Implicit bias is the phenomenon where a stereotype Search Committee Training influences a person’s decisions without his or her aware- The training consists of a thorough handbook and ness. Because the stereotype operates without a person’s a one-hour training, conducted by a faculty member awareness, it takes conscious effort to overcome this im- and an HR representative, to plicit bias [2]. There is a wide base of research literature make committee members documenting implicit bias and studying its effects on hir- conscious of biases and un- ing, including in academia. Social-psychology research has derstand ways to work to over- Everyone identified implicit bias based on a variety of stereotypes: come them in evaluation and gender bias, ethnic or racial bias, motherhood bias, sex- decision-making. The overar- holds uality bias, and disability bias have all been documented. ching message of this training unintentional is that an awareness of bias re- duces its effect. As committee biases members, simply reminding Elizabeth A. Burroughs is professor and head of the Department ourselves that we are prone that can of Mathematical Sciences at Montana State University. Her email to a bias can reduce the effect address is [email protected]. of the bias. influence our 1In 2012 Montana State received an NSF ADVANCE award (no. Search committees are en- 1208831) to increase the recruitment, retention, and advancement couraged to consider diversity evaluation. of women in STEM disciplines across campus. See the October as an asset and use it as one 7, 2016 Chronicle of Higher Education. In 2015, Maria Klawe of many criteria in evaluation. [1] issued a call in the Notices to talk about gender equality in The materials remind committees that anyone can demon- mathematics. strate a commitment to creating an inclusive workplace, For permission to reprint this article, please contact: and infusing diversity into the search process can help [email protected]. to hire a faculty member who values and respects differ- DOI: http://dx.doi.org/10.1090/noti1600 ence, no matter his or her identity. There is no expecta-

1304 Notices of the AMS Volume 64, Number 11 COMMUNICATION

tion that searches will hire an offer from us,” only undermine our efforts to recruit under-qualified applicants for diversity. Our obligation is to recruit broadly into the … awareness just to satisfy a diversity search pool, and then let the candidates decide if they of bias reduces goal. The search training want the position. directly addresses potential We advertise in publications and websites that target its effect. fears of search committees women and minorities. In the mathematical sciences, that their consideration of available options are the Association for Women in Math- diversity will somehow pre- ematics, the Benjamin Banneker Association, TODOS, the vent them from selecting National Association of Mathematicians, the Society for the “best” candidate by some objective measure, such as the Advancement of Chicanos/Hispanics and Native Amer- the number of publications or grant dollars. Many factors icans in Science, and the Caucus for Women in Statistics. usually prevent such a candidate from being deemed the We are also encouraged to use personal invitations to best—for­ example, if a candidate’s area of expertise over- encourage applications. Montana State challenges each laps too much with existing faculty members’ research, search committee member to personally contact ten poten- or if it doesn’t match a department’s existing research tial applicants with a “we’d love to review your materials” program enough. The training provides the advice that phone call or email. in cases where a department lacks diversity, a candidate who adds diversity may be the best candidate overall for Equity Advocate the program because of the benefit that diversity itself We do not rely exclusively on departmental committee affords. members to combat the effects of bias. The university has The final advice is that because stereotypes are mental initiated a program called the Equity Advocate program. shortcuts, one way to combat the bias from stereotypes Equity Advocates are faculty members from across cam- is to take time. Rushing through a review of materials is pus who have attended additional training to learn about more likely to activate stereotypes, so allow each appli- recognizing and preventing cation a careful review. In academic life, there are many bias. These individuals are competing demands, and it can be easy to try to minimize available to serve on search time on committee work. But search committee members committees in any depart- … one way must make a commitment to investing the time it takes ment. We have had two Equity to complete a search that won’t rely on quick judgments. to combat Advocates who are members Committee members can focus on whom to include rather of our department, but we than whom to exclude, and be sure to articulate a reason the bias from have also had Equity Advo- to advance or exclude someone. Making that decision cates from other departments stereotypes is public gives other committee members the opportunity serve on our search commit- to weigh in and serve as a check that the decision is not tees. Though every member of to take time. based in bias. the search committee attends Recruitment the search committee training, The key to being able to hire diverse candidates is which includes the materials about bias and acting against getting those diverse candidates to apply for the open po- bias, it is useful to have someone whose responsibility on sition. The strategies we use are focusing on the wording the committee is to recognize potential bias. of the advertisement, advertising in venues that appeal Equity Advocates can often speak up when a depart- to diverse candidates, and personally reaching out to mental colleague might not. Imagine being one of the few encourage candidates to apply. women in a department and serving on a search commit- The wording of the announcement should be carefully tee. These three underlying assumptions are unfair and considered, with a focus on broad descriptions of areas not necessarily true: rather than a narrow focus. This is based on research from (i) because you are a woman it is your job to recognize psychology: “On average, men apply for a job when they bias, meet only 60% of the qualifications. Women apply only (ii) it is your responsibility to alert the committee to when they meet 90% of them” [3]. In our department, we bias, and have focused on broad descriptions of research areas. We (iii) decisions you make are not based on bias. also include our desire for diversity in the position adver- There’s also an additional mental tax that comes from tisement, for example, by including language such as, “We not only fulfilling the usual obligations of the committee, hope to attract applicants who are committed to helping but also having to be attuned to everyone’s choices and students from diverse backgrounds succeed.” statements and screen them for potential bias. This is an The materials also caution against making assumptions unreasonable burden on women who serve on commit- about a candidate’s motivation or interest in an opening. tees—an already high-workload assignment. The practice Making inferences like, “She lives in a big city so she won’t of committee training for all, with the expectation that all be interested in our position,” or, “She will have offers at members of a committee will be alert for their own bias more prestigious universities so wouldn’t really consider and actively work against it, combined with the use of

December 2017 Notices of the AMS 1305 COMMUNICATION

Equity Advocates, whose only role can be to be alert for accommodate the partner. It is comparatively simpler to potential bias, results in a stronger committee structure. arrange for a non-tenure-track appointment. It is coop- eration and effort at the deans’ level, with partnerships Family Advocate Meetings during Interview across colleges, that make the system work. It doesn’t One component of the interview visits that has become always work, but it often does. standard on our campus is a visit with the University From a search committee member and department head Family Advocate. The Family Advocate is a tenured faculty perspective, the sooner the candidate declares the desire member who has been appointed to serve the university for an academic position for a partner, the easier it is to by disseminating information about resources and policies arrange and more likely it is to succeed. Early declaration affecting families on campus. perhaps flies in the face of advice many candidates have The Family Advocate has responsibilities for work with historically been given by their advisors and mentors, and current employees, but also participates in the recruitment many candidates often wait until the relative last minute process. She is completely separate from the search com- to seek an accommodation. We rely on our advertising and mittee, offers no feedback to the department or to the the Family Advocate program to assure candidates of our hiring committee, and is able to hold confidential meet- sincere desire to accommodate two-body opportunities. ings with every candidate. The meetings have a standard format: they last twenty to thirty minutes, and they open Changes at the Departmental Level with the Family Advocate describing the programs on These initiatives have had an effect at MSU. Since 2012, campus that support work-life balance, such as on-campus the university has hired 72 faculty members in STEM de- preschool, employee wellness benefits, and stop-the-clock partments. Of those 72 positions, 36 have been filled by tenure policies. There is always time for open question women. Our own department has made twelve hires at and answer, and it is often here that the candidate asks the assistant professor level since 2013, and six of those questions about dual-career partner accommodations or new faculty members are women, in contrast to only three asks questions about the campus climate. women among the thirteen tenured faculty. Our department has found these visits are useful as We’ve seen results beyond simply quantifying new a recruiting tool, but also as an honest reflection of our hires at the department level as well. One difference I department’s values. As a group, we intend to support our have noticed is that the search committee training gives faculty candidates, and it is a relief to know that they can the committee permission to discuss potential sources of have open and frank conversations early in the process bias. Having the search materials gives us a starting place and get advice and answers from someone knowledgeable for these conversations about bias. It has also given us a on campus. venue for discussions about diversity and the advantages I have occasionally served as a Family Advocate for that having students and colleagues with diverse perspec- candidates in searches from other departments or col- tives brings to our department. In the past, it has been leges, and I’ve seen how much candidates value this time. simple to lament our lack of diversity and blame the lack I’ve had conversations about schools for children, about of diversity within mathematics as a whole; now, we have opportunities for partner employment on campus and in knowledge and strategies that can help us to achieve a the region, and about quality of life in general in Bozeman. more diverse faculty. There is the secondary effect that we have an increased Dual-Career Arrangements awareness of the effects of bias in teaching. We had a de- Most everyone who has been involved in an academic partment-wide discussion of how gendered perceptions of search is aware of the two-body problem, where a candi- mathematicians affect individuals in our department. The date has a partner or spouse who is also looking for an immediate result was an awareness of how many women academic position. At Montana State, we are particularly receive anonymous comments on course evaluations that sensitive that while we live in a vibrant community with an refer to their appearance or contain sexual innuendos. It appealing natural setting, we are in a small city in a rural was a bit sad for everyone to realize that the women were state bordered by other rural states. We begin all searches not surprised to have had these experiences, and that the with the recognition that nearly all faculty candidates will men were surprised that women have these experiences. come with a spouse or partner, and that person will also This collective recognition of the gender bias in instru- need to find meaningful employment. We acknowledge ments has led to a discussion of how the department that we can’t leave the partners to fend for themselves in might deemphasize the use of this type of instrument, finding a new job, as might be reasonable in an institution and focus on better evaluation through more time and in a larger city. We also recognize that in the case that the care. Of course, the broader issue of how to address the partner is an academic, we have the opportunity to secure biased expectations that students bring to the classroom a dual-career couple. MSU has found that we can identify won’t be fixed by changing how we measure teaching a two-body opportunity rather than problem. effectiveness. Our department and the individuals in it The mechanics of how this works vary. Sometimes there continue to wrestle with issues of diversity as we consider is a tenure-track opening in a suitable department; other our responsibility to reduce the effects of bias. times we are authorized to pursue a target-of-opportu- Note: MSU maintains a two-page summary of search nity hire, where a tenure-track position is authorized to tips as well as the faculty-search toolkit, which is

1306 Notices of the AMS Volume 64, Number 11 COMMUNICATION available by request to the email address on the page www.montana.edu/nsfadvance. Both are influenced by the work of Fine and Handlesman [4]. The documents on the webpage are freely provided for your use after regis- tration. You can also purchase the toolkit from .

References [1] Maria Klawe, Time to talk about gender equality in math? Notices Amer. Math. Soc. 62 (2015). [2] Cheryl Staats and Charles Patton, “State of the Science: Implicit Bias Review 2013,” Kirwan Institute for the Study of Race and Ethnicity (2013). Retrieved from www.kirwan- institute.osu.edu/reports/2013/03_2013_SOTS-Im- plicit_Bias.pdf. [3] T.S. Mohr, Why women don’t apply for jobs unless they’re 100% qualified, Harvard Business Review (2014). [4] E. Fine and J. Handelsman, Searching for Excellence & Di- versity®: A Guide for Search Committees—National Edition. Women in Science and Engineering Leadership Institute, Uni- versity of Wisconsin—Madison (2012), wiseli.engr.wisc. edu/searchguidebooks.php .

ABOUT THE AUTHOR Elizabeth A. Burroughs has spent the past decade researching issues relevant to the teaching of K–12 mathematics. She has overseen fac- ulty searches as a committee chair and department head. In 2014–2015 she lived with her family in York, UK, while a Fulbright Scholar at the Elizabeth A. University of York. Burroughs

December 2017 Notices of the AMS 1307 OPINION Four Goals for Instructors Using Inquiry-Based Learning Chris Rasmussen, Karen Marrongelle, Oh Nam Kwon, and Angie Hodge

Communicated by Benjamin Braun

Note: The opinions expressed here are not necessarily those of Notices. Responses on the Notices webpage are invited.

In a recent Notices article [1], Ernst, Hodge, and Yoshi- a follow-up question nobu describe two pillars of inquiry-based learning (IBL): directed toward the deep engagement in mathematics and opportunities to expected response). What is the collaborate [2]. These two pillars identify key aspects of While such talk pat- instructor’s role what students do in an IBL classroom. What, however, is terns can be useful, the instructor’s role in a successful IBL classroom? This they do not usually in a successful question leads to a third pillar of IBL: instructor interest promote the pillars in and use of student thinking. In a stereotypical math of deep engagement IBL classroom? classroom, instructor interest is often expressed with in the mathematics prompts such as: Does anyone have a question? What and collaboration is the inverse of this linear transformation? How do you that characterize simplify this expression? These types of questions tend to student experiences with IBL. have an expected response and lead to what is referred to In this article, we discuss ways in which instructors as an Initiation-Response-Evaluation pattern of instructor can achieve four research-based goals in IBL classrooms. and student talk. The instructor initiates a discussion by These goals and corresponding instructor actions are asking a question, a student responds, and the instructor grounded in research conducted in multiple inquiry-based indicates whether the response is correct or not (or asks classrooms, including differential equations and linear algebra [3], [4], [5]: Chris Rasmussen is professor of mathematics education at San Diego State University. His email address is crasmussen@mail (1) get students to share their thinking, .sdsu.edu . (2) help students to orient to and engage in others’ Karen Marrongelle is dean of the College of Liberal Arts and Sci- thinking, ences and professor of mathematics and statistics at Portland State (3) help students deepen their thinking, and University. Her email address is [email protected] . (4) build on and extend student ideas. Oh Nam Kwon is professor of mathematics education at Seoul As anyone who has tried to implement IBL can attest, National University. Her email address is [email protected] . realizing these goals takes time, patience, occasional Angie Hodge is assistant professor of mathematics at Northern failures, and the willingness to try out new patterns of University. Her email address is [email protected] . interaction with students (often called “talk moves”). For permission to reprint this article, please contact: While these four goals are not an exclusive list of instruc- [email protected]. tor responsibilities, they serve as a solid foundation for DOI: http://dx.doi.org/10.1090/noti1597 successful teaching in IBL classrooms.

1308 Notices of the AMS Volume 64, Number 11 Opinion

#1: Get Students to Share Their Thinking we do not normally use in everyday conversation, yet talk In an IBL classroom, students spend their time actively moves such as the ones below can be quite effective in working on problems (often with their peers) that are promoting student discourse: novel and/or challenging to them. Instructors strive to •• “Who can repeat in their own words what Juan just bring forth student ideas and make them public, even if said?” the answers are incorrect or the students are tentative •• “Do you agree with what Darrel just said?” about their work. Creating a classroom environment where •• “Say that again please so that everyone can hear.” students feel safe to express their thinking is a challenge •• “Can someone rephrase Diane’s explanation in their and not always easy to achieve, but research grounded own words?” in practice has revealed various prompts that can help •• “So, Debbie, is Minsoo saying that…?” realize this goal: •• “Jesika, what do you think about what Ding just said?” •• “Dave, I know you haven’t finished the problem, but These types of talk moves encourage other students tell us your initial thinking.” to revoice the thinking of their classmates. Sometimes •• “Take your time; we’re not in a rush.” student revoicing leads to a different conclusion or jus- •• “Can you say more about that?” tification and thus creates an opportunity for a broader •• “That’s an important point. Keisha, can you say that debate among multiple students. A student misinterpre- again so that everyone can hear?” tation may come to the surface, encouraging the original •• “Learning from mistakes is an important part of doing speaker to use more precise mathematical language or mathematics. Who can share with us an initial approach explain their idea in a different way. All of these outcomes that didn’t work out?” are mathematically productive. These and similar talk moves (Figure 1) are productive for eliciting student thinking because they clearly com- #3: Help Students Deepen Their Thinking municate the expectation that students share their ideas Even if students express their thoughts and listen to rather than be completely correct. others’ ideas, the discussion can still fail to be mathemat- ically productive if it does not include solid and sustained reasoning. Most students are not skilled at pushing to Can you say more about that? deepen their own reasoning or that of their classmates. Therefore, a key role of the instructor is to continuously and skillfully press students for reasoning, explanation, and justification. Instructors can help achieve this by using the following prompts: •• “How can we check to make sure this is correct?” •• “What is the reasoning that allows you to make that conclusion?” •• “Do you agree or disagree and why?” •• “How does that relate to what we learned yesterday?” •• “Can anyone come up with a different way to explain that?” •• “OK, I hear what you are saying, but what about this counterexample?” Figure 1. “Talk moves” elicit student thinking. •• “We now have three different solutions and not all three can be correct. Work with your partner(s) to decide, with But getting students to talk is, of course, just the start. justification, whether each solution is correct or not.” More often than not, students’ initial thinking is muddled As these questions suggest, IBL instructors must take a or difficult to interpret. Rather than evaluating the re- proactive role in pressing students to develop their ideas sponse or asking a leading follow-up question, an instruc- and to explain and justify their thinking, navigating be- tor might choose to repeat back what the student said, tween students’ ways of reasoning and conventions of the with or without an interpretation or rephrasing, and then broader mathematical community (Figure 2). check back with the speaker for accuracy. This type of talk move is referred to as revoicing and is a highly effective #4: Build On and Extend Students’ Ideas tool for initiating classroom discussions, in part because IBL classroom discussions are opportunities for stu- it conveys no evaluation of students’ initial contributions. dents to report on the progress they have made working (individually, in pairs, or in small groups) on challenging #2: Help Students Orient To and Engage With problems. Such occasions are also opportunities for the Others’ Thinking instructor to build on student work to advance their math- Once one or more students have shared their initial ematical agenda, to make connections to more formal or thinking, the role of the IBL instructor is to promote cross- conventional mathematics, and/or to help students see talk among students where they attend to and make sense and appreciate that they are actually doing mathematics. of other students’ thinking. Instructor prompts that can Using the following instructor moves are some ways in help achieve this goal often make use of questions that which instructors can build on and extend student work:

December 2017 Notices of the AMS 1309 Opinion

References [1] D. C. Ernst, A. Hodge, and S. Yoshinobu. What is inqui- ry-based learning? Notices Amer. Math. Soc. 64 (2017), no. 6, 570–574. [2] S. Lauren, M-L. Hassi, M. Kogan, A-B. Hunter, and T. Weston, Evaluation of the IBL Mathematics Project: Student and In- structor Outcomes of Inquiry-Based Learning in College Math- ematics, www.colorado.edu/eer/research/documents/ IBLmathReportALL_050211.pdf [3] K. Marrongelle and C. Rasmussen. Meeting new teaching challenges: Teaching strategies that mediate between all lecture and all student discovery. In M. Carlson and C. Ras- mussen (eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education, Math. Assn. Amer., 2008, pp. 167–178. Washington, DC. [4] C. Rasmussen and O. Kwon. An inquiry oriented approach to undergraduate mathematics, J. Math. Behavior 26 (2007), Figure 2. After being prompted to consider a 189–194. classmate’s claim, students work in small groups to [5] M. Wawro, M. Zandieh, C. Rasmussen, and C. An- decide if they agree or disagree and why. drews-Larsen. The Inquiry Oriented Linear Algebra Project, 2017, iola.math.vt.edu •• Restating what students said or did in more conven- tional or formal terms. Photo Credits •• Introducing a new but related concept, definition, repre- Figure 1 courtesy of Chris Rasmussen. sentation, or procedure that extends what students did. Figure 2 courtesy of the University of Nebraska at Omaha. •• Restating a student’s explanation and attributing au- Photo of Chris Rasmussen courtesy of Chris Rasmussen. Photo of Karen Marrongelle courtesy of Karen Marrongelle. thorship to the student or students, i.e., creating the Photo of Oh Nam Kwon courtesy of Oh Nam Kwon. sense that mathematics is arising out of students’ own Photo of Angie Hodge courtesy of Stan Yoshinobu. work. •• Restating student ideas in ways that connect to estab- lished mathematical culture. Benefits of such instructor moves include highlight- ing how student ideas fit into the larger mathematical trajectory; providing scaffolding for students to clarify, to elaborate, and to extend their mathematical positions; representing mathematics as something that can develop from students’ own work; and supporting students’ en- culturation into the discipline of mathematics.

Conclusion Effectively supporting students’ deep engagement in mathematics and peer-to-peer collaboration is an omni- present challenge for instructors. The four goals discussed here provide some insight into what an instructor can do to realize the student-focused pillars of deep engagement in mathematics and collaboration. Just as a stool with three legs is sturdier than one with only two, IBL class- rooms are on a stronger foundation when all three pillars are taken into account: the two focused on the activity of students and the one focused on the activity of the instructor.

1310 Notices of the AMS Volume 64, Number 11 Opinion

A MERICAN M ATHEMATICAL S OCIETY

ABOUT THE AUTHORS Chris Rasmussen is currently study- ing departmental change and factors that influence student success over the entire progression of the intro- ductory mathematics courses. When not doing research or teaching math, Chris enjoys traveling, playing ten- nis, and going to a dog beach with Chris Rasmussen his two Basenjis. Karen Marrongelle, when she’s not popping into classrooms around campus, enjoys skiing and running in the mountains of the Pacific Northwest and cooking food from her garden.

Karen Marrongelle

Oh Nam Kwon, when she is not teaching or researching with her graduate students, enjoys dancing, singing with the SNU Faculty Choir, and golfing with her husband.

Oh Nam Kwon

Angie Hodge is co-director of the Academy of Inquiry Based Learning. She has taught a variety of courses using IBL over the last ten years. She travels all over the world to run ultramarathons, including the Leadville 100-mile run.

Angie Hodge

www.ams.org/ students

December 2017 Notices of the AMS 1311

COMMUNICATION

Dorothea Rockburne and Max Dehn at Black Mountain College

David Peifer

Communicated by Thomas Garrity

ABSTRACT. The artist Dorothea Rockburne was in- spired by the mathematician Max Dehn while a student at Black Mountain College.

Dorothea Rockburne is a New York based artist whose paintings are inspired by her fascination with mathe- matics. Over her more than fifty years as a successful exhibiting artist, Rockburne has won many awards for her work, including the 1999 American Academy of Arts and Letters Jimmy Ernst Lifetime Achievement Award in Figure 1. Dorothea Rockburne is a distinguished Art and the 2009 National Academy Museum and School artist with a fascination for mathematics. Rockburne of Fine Arts Lifetime Achievement Award. A major retro- in 2014, photo by Zia O’Hara, (on left) and in her New spective, “Dorothea Rockburne: In My Mind’s Eye,” held in York studio in 2009 (on right). 2012 at the Parrish Art Museum, ex- hibited works spanning her career, an experimental liberal arts college including works from her Pascal with an emphasis on the arts (see series and paintings inspired by her the ability of sidebar). reading of Poincaré. A multi-gallery In 1945, the BMC faculty was exhibition at the Museum of Mod- mathematics to shed joined by the German refugee math- ern Art in 2014, Dorothea Rock- ematician Max Dehn. Dehn taught burne: Drawing Which Makes Itself, light on the underlying mathematics, philosophy, Greek, included some of her earliest works principles of nature and Latin. He and his wife lived on from her series. the top floor in a small house on The catalyst of Rockburne’s in- campus. Like most of the faculty, terest in mathematics came during the Dehns ate their meals in the the first two years of her undergraduate education. In dining hall and participated in the social life on campus. 1950, at age 17, Rockburne left her home in Montreal, On Sundays, when meals were not provided, Dehn and his Canada, to study painting at Black Mountain College (BMC), wife frequently invited students to their home for dinner. Dehn was well known for taking faculty, students, and visitors on long hikes into the surrounding mountains in search of local wildflowers. David Peifer is professor of mathematics at the University of North Before arriving at BMC, Rockburne already had a solid Carolina at Asheville. His email address is [email protected] . education in painting. While she was an elementary stu- For permission to reprint this article, please contact: dent, she won a merit scholarship to the Ecole des Beaux- [email protected]. Arts in Montreal, for Saturday classes. Rockburne always DOI: http://dx.doi.org/10.1090/noti1599 had a deep respect and fascination with nature. She recalls

December 2017 Notices of the AMS 1313 COMMUNICATION this wonder for nature being sparked one day during an nature and who would show her how mathematics could outdoor painting class on Mount Royal, in Montreal: be used to unlock its guiding principles. She recalls a steep hike, about a mile uphill behind the Dehns' home, Although my teacher praised my painting, as where they walked to a small waterfall. Dehn discussed the I stood and looked first at my work and then golden ratio and “showed [her] that every kind of growth at the landscape, I observed and questioned, has recognizable mathematical properties.” Dehn recom- “Nature does it better. How?” More specifically, mended that she read several books, including Science and I asked myself, what are nature’s organizing Method by Henri Poincaré, The Laws of Thought by Henry principles? These questions probably arose Parker Manning, The Fourth Dimension by C. Howard because of my father’s Algonquin traditions: Hilton, and Flatland by Edwin A. Abbott. In later years, “Nature is sacred and should be understood, Rockburne would go back to these books for inspiration. not violated.” This moment of inquiry would During Rockburne’s second year at BMC, Dehn passed later directly lead to my ongoing passion for away. Rockburne married Carroll Williams, an apprentice mathematics as the basis for the understand- teacher and former student at BMC. She participated in ing of nature. John Cage’s Theatre Piece No. During high school, Rockburne won another schol- 1, sometimes called “The First arship to take classes at the Montreal Museum School. Happening.” She attended the Gordon Webber introduced Rockburne to artistic theories international pottery seminar not afraid from the Bauhaus and Russian constructivism. Rockburne lead by the Japanese potter recalls that this was “a different kind of art thinking and Shoji Hamada. She danced to pick was a large part of what drew me to BMC.” with Merce Cunningham. In up a real In her first week at BMC, Rockburne tried taking a math- 1954, the family moved to ematics class with Dehn, but quickly found she was not Manhattan. Unfortunately, the mathematics prepared and dropped out. She says that Dehn must have marriage did not last long. seen something in her mathematical abilities, because a Soon Rockburne was a young text to see few days later he offered to teach her mathematics as it is single mother working hard to seen in nature. He suggested that she accompany him on bring up her daughter in New what she his morning hike. Rockburne was greatly influenced by her York City. time with Dehn and this would ultimately shape her career Rockburne stayed in touch could get out as an artist. Rockburne fondly recalls her time with Dehn: with many BMC artists, stu- of it dents, and faculty who had Dehn considered my lack of math instruction moved to the New York area. an advantage. I wasn’t mathphobic, and, as he Encouraged by her BMC so happily put it, I hadn’t been poisoned. Dehn friends, in 1957 Rockburne had an exhibition of her work had a lively, disciplined, but fearless mind. at the Nonagon Gallery in New York. The show got good His enthusiasm for everything was infectious. reviews and she sold several works. Rockburne herself When I told him I was having some difficulties was encouraged, but she felt she needed more time to with assignments, he said, “What you need is develop. She recalls, “I knew that these paintings did not to understand the principles of math as they emotionally or creatively correspond to my inner need or occur in nature. Why don’t you join me every voice. … I returned to my studio for ten years to work, day on my early morning hike, and I will teach study, and form a voice.” By keeping her expenses low, she you mathematics for artists through nature?” managed to support herself and care for her daughter. For In Dehn Rockburne found a man who, like her father, a short while she worked at the Metropolitan Museum of had a great respect for nature, someone who understood Art and assisted in cataloguing the museum’s collection of Egyptian antiquities. Most of the time she worked as a waitress. “To stay sane,” she went back and reread the books that Dehn had suggested. She also began her own investigations into mathematics such as reading books on non-Euclidean geometry and set theory. She was not afraid to pick up a real mathematics text to see what she could get out of it. She read these books with the goal to “use them as a learning foundation from which to conceive art.” She was fascinated with the history of mathematics and said, “By understanding the history of mathematics, I learned of an exquisite emotional beauty of thought. This in turn gave me greater access to an understanding of a more universal creative process.…” Dehn had always Figure 2. Max Dehn greatly influenced Dorothea stressed the importance of history. If Dehn had talked Rockburne (1952) at Black Mountain College. with Rockburne about hyperbolic geometry, which he used

1314 Notices of the AMS Volume 64, Number 11 COMMUNICATION

Max Dehn Max Dehn received his doctorate under David Hilbert in Göttingen in 1900, the same year that Hilbert gave his famous lecture outlining 23 problems for the twentieth century. The next year Dehn was the first to solve one of the problems on Hilbert’s list. He solved the third prob- lem, showing that the Archimedean axiom is logically needed to prove that tetrahedra of equal base and height have equal volume. Dehn went on to do ground-break- ing work in topology and the theory of infinite groups. In 1907, Dehn coauthored with the first comprehensive articles on topology, including the first complete proof of the classification of 2-manifolds. Dehn’s work on 3-manifolds included inventing a way to construct new 3-manifolds from existing ones. This process is now called Dehn surgery and is still important in topology today. Dehn used his surgery to construct an infinite set of examples of Poincaré homology spheres. After escaping the Nazis, Max and his wife Toni Dehn Before this, only Poincaré’s single example was known. eventually settled at BMC in the mountains of North Dehn was the first to formulate the word and conjugacy Carolina. problems for finitely presented groups. In an amazing paper from 1911, using hyperbolic geometry, Dehn was love for hiking and during his life developed an extensive able to solve these problems for surface groups. The knowledge of botany and the flora of the forest. He had, word and conjugacy problems have continued to play what we would call today, an interdisciplinary approach. an important role in the study of infinite group theory André Weil, in his autobiography, The Apprenticeship of through present day. In a 1914 paper, Dehn was the a Mathematician, compared Max Dehn to Socrates and first to prove that the right and left trefoil knots are not said, “...for such a man, truth is all one, and mathematics equivalent. Later in his career he worked on the mapping is but one of the mirrors in which it is reflected. … [Dehn class groups, finding a finite set of generators now known was a] humanistic mathematician, who saw mathematics as Dehn twists. Dehn was an intuitive geometer at heart. as one chapter—certainly not the least important—in the Many of the mathematical ideas attributed to him, such history of the human mind.” as Dehn diagrams, Dehn surgery, and Dehn twists, are Dehn had been forced by the Nazis to leave his posi- in fact simple diagrams that are easy to visualize. Their beauty is that these simple diagrams provide a way to tion at the University of Frankfurt in 1935. Three years understand more general and complicated situations. later, he was arrested the day after Kristallnacht. Dehn Dehn seems to have been admired and respected by and his wife went into hiding and eventually escaped everyone he met. He had grown up with an interest in the to , where Dehn took a position at the Technical arts and music as well as science and mathematics. He University in Trondheim. After the Germans invaded learned to read and speak several languages, including Norway, the Dehns again had to go into hiding and even- Greek, Latin, French, Norwegian, and English. He had tually escaped to . There they began their long a lifelong interest in philosophy, in particular, ancient travels to the US through Siberia and Japan and across Greek and modern German philosophy. He had a great the Pacific until eventually settling at BMC.1 so frequently in his work, or if he talked to her about his “A, C and D (from Group/And),” a wall-size installation views on set theory, axiomatic geometry, topology, and of paper and chipboard with natural pigments, rolled the general abstraction of mathematics, details are lost out on the wall and floor, and held together by nails. In to time. It seems clear that at least Dehn had sparked March of 1972, Artforum featured her work “Domain of her interest in some of these ideas. In particular, as a The Variable,” on its cover and included an article and an combinatorial topologist, he could not have helped but interview with Rockburne. to provide her with some of his insights concerning the Rockburne’s earliest works were identified with the shape of space. minimalists. The 1971 work “Scalar” (Figure 3) is inspired In the early 1970s Rockburne did a series of works by her study of linear algebra. It is made of several 30”x40” based on her reading of Cantor and set theory, which sheets of chipboard that had been saturated in crude oil, propelled her into the forefront of the art world. These stacked in her loft, and then left to sit for a summer. These works, including “Set,” “Intersection,” and “Null Set,” sheets were then attached to the wall by nails in an over- were shown in her first two solo exhibits, as well as group 1See the 2002 Notices article on “Max Dehn, Kurt Gödel, and the exhibitions in New York, Minneapolis, and Eindhoven in Trans-Siberian Escape Route” www.ams.org/notices/200209/ The . The purchased fea-dawson.pdf .

December 2017 Notices of the AMS 1315 COMMUNICATION

Rockburne’s 1973 “Drawing Which Makes Itself: Neigh- borhood” (Figure 4) is a wonderful example of her ideas to change the way space and time are viewed in a work of art. This work consists of a large rectangular sheet of translucent vellum that has creases folded from corner to corner. This is placed horizontally at the exact center of the wall. There are several lines drawn with different sized colored pencils scattered on the wall around and under- neath the vellum. These lines clearly indicate the position of the vellum if it were carefully folded and unfolded to move geometrically along the wall. While the placement of the lines seems to follows some mathematical laws, their position does not look symmetric or mechanical. The color and size of the lines seems to indicate a rhythm and passing of time. Standing back, one can almost see the sheet of vellum beautifully dancing across the wall and

© 2017 Dorothea Rockburne / Artists Rights Society (ARS), New York leaving only a trace of its passing. Figure 3. Rockburne’s 1971 “Scalar,” inspired by linear Rockburne’s use of lines on the wall around the vellum algebra, had an intriguing three-dimensional aspect. clearly indicate a break from the tradition of drawing and Museum of Modern Art. painting made on a rectangular flat surface. Her works lapping array. The whole work measures approximately move across the wall and roll out onto the floor. These 10’x6’. In the art world, it is praised as an early example of drawings have broken free of the rectangular frames that experimentation with materials. Rockburne says she used hold bounded most other drawings. In this way, her work crude oil because she was poor at the time and couldn’t has opened up a new dimension. It is also interesting to afford paints. What she really loved about the oil was how note that in this work she uses translucent vellum. This it worked its way through the board and how the image provides the ability to see through and around the mate- on one sheet was superimposed on the sheets above and rial. Just as in other works where she rolls or folds paper, below. It was this three-dimensional aspect that intrigued she is showing both sides of the paper. This gives the her. By sitting for a summer, the oil had transformed the viewer a multi-layered perspective. sheets. Since the sheets had been stacked, they showed a It is interesting to note what Rockburne means by the dynamic of this slow transformation of oil soaking down title “Neighborhood.” As a mathematician, familiar with through the sheets and imprinting on each other. The Rockburne’s references to topology, I would be inclined placement of the sheets on the wall provides the viewer to think of a topological neighborhood. However, this is with Rockburne’s interpretation of this dynamic system. not the case. The title comes from her reading of Flatland Rockburne became good friends with Sol LeWitt and and the mention that one-dimensional Linelanders could other minimalists. She had several works in group exhi- never pass each other. Thus, neighborhoods for them bitions featuring minimalist artists. However, from the are like marriages. This gives an even better insight into start, Rockburne did not feel that her art fit within the Rockburne’s thinking. Flatland is about trying to visualize framework of minimalism. “Like them, my work was not decorative nor did I make decisions for aesthetic reasons. My art-making decisions were based on the principles of topology, a complex study of continuous space. … This is a different spatial concept from the minimalist practice of working on a flat grid.” Rockburne had studied the ways in which artists throughout history created space in their works. She had studied ancient Egyptian art, where small fragments can contain incredibly large scale. Early on she had learned of the Renaissance artists and their use of geometry, vanishing points, and the golden mean. She understood the significance of the rebellious Dada artists and how these ideas, mixed with Freud’s theories of the subconscious, went on to influence surrealism and abstract expressionism giving a more dreamlike sense of space. Rockburne understood how the cubists had defined a new fragmented canvas that allowed for views from

varying angles and staggered moments of time. With her © 2017 Dorothea Rockburne / Artists Rights Society (ARS), New York work, she wanted to define a new way to view space that Figure 4. Rockburne’s 1973 “Neighborhood” provided incorporated and drew from these past perspectives, but a new way to view space and time. Museum of Modern also would go beyond to new dimensions. Art.

1316 Notices of the AMS Volume 64, Number 11 COMMUNICATION

of natural tones or pure white paper. Many of the deep blues and reds and the gold are borrowed from specific

paintings by Renaissance masters. Gold leaf was used in Renaissance paintings because of its wonderful ability to reflect natural light and to a create special brilliance, as in halos. In many of these older works there is subtle ge- ometry in the composition, sometimes from the gestures or lines of sight of people in the paintings. Other times shading and colors create planes and simple geometric shapes. The lines and planes in Rockburne’s works are inspired by the subtle geometry in these masterpieces. The 1530 painting “Visitation,” by Pontormo, was an in- spiration for many of the Pascal works. Conclusion. In some ways Rockburne’s career is com- © 2017 Dorothea Rockburne / Artists Rights Society (ARS), New York . Figure 5. Rockburne’s Pascal works, such as “Pascal, pletely different from Dehn’s. She is an artist and Dehn State of Grace” (left, 1986–87), were inspired by the was a mathematician. Her career was only beginning in philosophy of Blaise Pascal and “Visitation” (right, 1950 while Dehn’s was coming to an end. However, in 1528–29) by Jacopo da Pontormo. Private collection. many ways the lifelong endeavors of these individuals run Pieve di San Michele Arcangelo, Carmignano. parallel. Both have shown a deep intuitive sense of geom- etry. Both have worked to improve the understanding of different dimensions, just as Rockburne is trying to do geometry and space in their own disciplines. Dehn’s math- in her work. Also in Flatland women are line segments. ematics shows an astonishing geometric insight. His early Once I made this connection, I could not help but see this works, including his solution to Hilbert’s third problem, work as a woman (a 1-dimensional being), dancing in time answered deep philosophic questions about Euclidean through Flatland (a 2-dimensional world), with an order geometry. His solution to the word problem for surface induced by simple mathematical folds, translations and groups showed a fundamental connection between group rotations in space (a 3-dimensional universe). The analogy theory and hyperbolic geometry. Dehn’s work on three goes even further if you consider that Abbot’s book is a manifolds and knot theory was decades ahead of its time. satire on 19th century society. The women in Flatland are Dehn’s geometric insight has had consequences reaching line segments and thus at the bottom of the social stratus, back to Euclid and forward to recent work in geometric whereas, in 1970 Rockburne is a woman struggling to be group theory and low dimensional topology. part of a male-dominated profession. This work becomes Rockburne’s works show similar geometric insight, but a self-portrait. Whatever Rockburne intended us to feel applied to paintings and drawings. Like Dehn, Rockburne and think when we see this work, there is no question is deeply interested in history and realizes the importance that she is extending the way in which space and time are of studying the works of the past. Her education was in the represented in paintings and drawings. classics, as well as the most avant-garde of contemporary Art historians have found it is hard to classify Rock- art. She has a deep understanding of space, shape, and burne’s work. This is partly because, in her 50-year career, perspective. She has the mind of a geometer. Her early she has never stayed with one style. Her earliest work works in the 1970s were multi-dimensional, using the is almost monochromatic. Her later work is full of rich whole room to communicate with the viewers and draw colors, including deep blues and bright golds inspired by them into the experience. Her works play with geometric her study of Renaissance paintings. She has paintings that shapes, time, and space in ways that other artists had measure as small as 4”x6” and others that cover walls over never done before. It is hard to know exactly what specific 40 feet high. She has painted on chipboard, paper, vellum, mathematical details Rockburne learned from her mentor parchment, and copper. Her painting techniques include Max Dehn, almost seventy years ago. However, it is clear classic styles of oil painting and frescoes, but also include that theirs was the meeting of two great geometric minds. unique applications of watercolors or crude oil. In general art historians call her a post-modern painter. In 1988, Rockburne first showed her “Pascal” works (Figure 5), inspired by the philosophy of Blaise Pascal. Here she uses stretched canvas on rectangular (usually golden rectangles) and triangular frames. These are then overlapped to create a painting that is no longer rectan- gular, but polygonal. The canvases are painted to suggest overlapping and transparency, in a way that is not always consistent with reality. This causes a kind of folding and turning of space that makes it appear that the paintings are looking back at themselves. She uses deep and brilliant colors in these works, a bold step from her earlier use

December 2017 Notices of the AMS 1317 COMMUNICATION

Black Mountain College Opened in 1933 and located in a remote area of the North Carolina mountains near Asheville, Black Mountain College (BMC) was an experimental liberal arts college with an emphasis on the arts. Although BMC only ex- isted for 24 years, the college faculty and students have had a tremendous influence on contemporary painting, dance, music, poetry, ceramics, performance art, and architecture. The curriculum at Black Mountain College was a mix of John Dewey’s philosophy on education and theories from the Bauhaus, a German school of design. The production of art—paintings, theater, writing, weaving, furniture making—served as a central core in the students’ edu- cation. The college was run under democratic principles The Studies Building on the Lake Eden campus of with no administration, no grades, and no accreditation. Black Mountain College. Faculty owned the campus. Decisions were made by Asawa and Kenneth Snelson, and poets Robert Creeley consensus. Campus upkeep was done by the community and M. C. (Mary Caroline) Richards. of faculty and students. This included building barns, The campus also attracted notable faculty in the sci- homes, and at one point, a large Studies Building. In 1933 ences. Both Nathan Rosen and Peter Bergmann taught the BMC founders hired Josef and Anni Albers, who had at BMC for a year after doing postdoctorial work with been teaching at the Bauhaus before it was closed by Albert Einstein. the Nazis. Partly due to the Albers’ influence, the college Einstein visited the campus one day in 1941 and re- soon began to attract refugee painters, composers, and mained an “honorary member” of the advisory board for other artists from Germany and across Europe. Over several years. Natasha Goldowski Renner, a metallurgist, the years, the college became a magnet for avant-garde had been one of the top-ranking woman in the Manhattan European and American artists. The list of distinguished Project before teaching at BMC for six years. In 1949, painters who were faculty or students at BMC includes Buckminster Fuller built his first successful geodesic African-American Jacob Lawrence, Willem de Kooning dome on the campus. and of the “New York School,” and Two of Dehn’s BMC students went on to get PhDs in pre-pop artist . A sample of artists mathematics and statistics. Several mathematicians vis- from other disciplines includes composer-choreographer ited Dehn at BMC, including Emil Artin, Wilhelm Magnus, duo John Cage and Merce Cunningham, sculptors Ruth Carl Siegel, and André Weil.

References 1) M.E. Harris, The Arts at Black Mountain College, The MIT Press, Cambridge MA, 1987. ABOUT THE AUTHOR 2) A. G. Longwell, Dorothea Rockburne: In My Mind’s Eye, The David Peifer has been a board mem- Parrish Art Museum, Southampton, NY, Distributed Art Pub- ber at the Black Mountain College lishers, New York, NY, 2011. Museum and Art Center for the 3) R. B. Sher, Max Dehn and Black Mountain College, Math. Intell., 16 (1994). past nine years. He enjoys hiking and mountain biking with his family Image Credits and friends. Figures 1, 3, 4, and 5 (left) courtesy of Dorothea Rockburne. Figure 2 (left by Trude Guermonprez; right by Marie Tavroges David Peifer Stilkind) courtesy of the Western Regional Archives, State Archives of North Carolina. Figure 5 (right) “Visitation” from Scala / Art Resource, NY. Photo of Max and Toni Dehn by Trude Guermonprez, courtesy of the Western Regional Archives, State Archives of North Carolina. Photo of BMC Studies Building by Hazel Larsen Archer, “The Path to the Studies Building, Black Mountain College,” ca. late 1940s–early 1950s, courtesy of Estate of Hazel Larsen Archer and Black Mountain College Museum + Arts Center. Author photo by Brenda Carleton.

1318 Notices of the AMS Volume 64, Number 11 COMMUNICATION

Autobiography of Wentsun Wu (1919–2017)

In 1946 I met the great geometer Chern Shiing-Shen. Chern is particularly renowned for the introduction of CHERN Classes and CHERN numbers of unitary bundles which are of extreme importance among the various kinds of characteristic classes of fiber bundles. At that time Chern was in charge of the newly established Institute of Mathematics belonging to Academia Sinica. This meeting with Chern was decisive for the future of my career in mathematics. Chern admitted me as one of the young students in his institute, all learning under his guid- ance. One year later I brought out my first paper about a simple proof of the product formula of sphere bundles discovered by H. Whitney for which his original proof was extremely complicated and had never been published. In 1946 I also passed the national examination for Wentsun Wu won the 2006 in Mathematical sending students abroad, and in 1947 I was sent to study Sciences for his contributions to the new mathematics in France as part of a Sino-France Exchange interdisciplinary field of mathematics mechanization. Program. I went to Strasbourg to study under Professor Ch. Ehresmann. In 1949 I passed my doctor thesis and EDITOR’S NOTE. The Shaw Prize Foundation kindly then went to Paris to study under Professor H. Cartan. provided this autobiography of Wu from their website During my stay in Strasbourg I made the acquaintance of shawprize.org , which recognizes Wu’s “outstanding R. Thom who was also a student of Cartan but who while contributions [to] topology, automatic reasoning, ma- at Strasbourg, had much contact with Ehresmann too. chine proof, algebraic geometry, Chinese mathematics The collaboration was a very fruitful one. In 1950 Thom history, ,” including the “Wu formula,” “Wu discovered the topological invariance of Stiefel–Whitney Class,” and “Wu method.” According to the prize cita- classes, while I, with the aid of Cartan, discovered the tion, “Wu introduced a powerful mechanical method classes and formulas now bearing my name. that transforms a problem in elementary geometry into In 1951 I returned to China, and in 1953 became a re- an algebraic statement which lends itself to effective searcher in the Chinese Academy of Sciences (CAS) where computation. This method of Wu completely revolu- I remain to the present day. From 1953 onwards I made a tionized the field, effectively provoking a paradigm somewhat systematic investigation of classical topological shift.” but non-homotopic problems which were being ignored at that time owing to the rapid development of homotopy I was born in Shanghai, China, on May 12, 1919. I re- theory. I introduced the notion of imbedding classes, and ceived my BS degree in mathematics from Jiaotung Uni- established a theory of imbedding, immersion, and isotopy versity, Shanghai, in 1940 during the war against Japan of polyhedra in Euclidean spaces which was published in (1937–1945). On graduating from university, because of book form later in 1965. In 1965, I was awarded one of the war, I had to teach for years in junior middle schools the three national first prizes for natural sciences for my bringing to a halt my further learning of mathematics. work on characteristic classes and imbedding classes. During the cultural revolution I was sent to a factory For permission to reprint this article, please contact: manufacturing computers. I was initially struck by the [email protected]. power of the computer. I was also devoted to the study DOI: http://dx.doi.org/10.1090/noti1604 of Chinese ancient mathematics and began to understand

December 2017 Notices of the AMS 1319 COMMUNICATION Advertise in the what Chinese ancient mathematics really was. I was greatly struck by the depth and powerfulness of its thought and its methods. It was under such influence that I in- vestigated the possibility of proving geometry theorems in a mechanical way. In 1977 I ultimately succeeded in developing a method of proving mechanical geometry Notices theorems. This method has been applied to prove or even of the American Mathematical Society discover hundreds of non-trivial difficult theorems in elementary geometries on a computer in a simplistic way and was henceforth called WU’s method in the literature. The discovery of WU’s method marks the second turning point in my scientific life, the first one being my meeting with Chern. Since that time I have completely changed my direction of research and concentrated my efforts on ex- tending the method in various directions, both theoretical and practical, aiming at what I have called “mechanization of mathematics.” Among the honors I have received for my research we may cite: In 1991 I received the mathematics award from and became a member of the Academy of Sciences for the Developing World (previously called The Third World Academy of Sciences). In 1997 I received the Herbrand Award on automated deduction for my mechanical ge- ometry theorem-proving. In 2001 I was awarded the first State Supreme Science and Technology Award of the Chinese government in recognition of my achievements in mathematics research, both in pure mathematics and in mathematics mechanization. Connect with an audience of approx- Finally, Mumford and I together were named as winners imately 30,000 subscribers through of the 2006 Shaw Prize in Mathematical Sciences for our Classified and Display Advertising research in pure mathematics, especially with regard to computer applications to mathematics which represents in the Notices of the American Mathe- a new role model for mathematicians of the future. matical Society, available in print and online. 12 September 2006, Hong Kong. As the world’s most widely read mag- Photo Credit Photo of Wentsun Wu courtesy of the Shaw Prize Foundation. azine aimed at professional mathe- maticians, the Notices is an excellent medium for announcing publica- tions, products, and services, and for recruiting mathematical scientists in academic, industry, and government.

Learn more about advertising in the Notices at www.ams.org/noticesadvertising.

1320 Notices of the AMS Volume 64, Number 11

BOOK REVIEW

The Real and the Complex A Review by Judith V. Grabiner

The Real and the Complex: A History of Analysis in occurred only at a finite number of identifiable isolated the 19th Century points. Most mathematicians of the eighteenth century Jeremy Gray (Lagrange was an exception) regarded questions like Springer, 2015 “what is a limit?” or “how can you talk about infinity or Paperback, 350 pages the infinitely small?” as perhaps of philosophical interest ISBN–13: 978-3-319-23714-5 but not essential to mathematical eBook ISBN: 978-3-319-23715-2 progress. Lagrange understand- ably, though mistakenly, believed Most basic formulas and techniques in what is often called that he had given the calculus a takes us on advanced calculus were developed in the eighteenth cen- rigorous foundation based on tury. More careful definitions and more rigorous proofs, what he thought of as the algebra new results that are too subtle to do without such proofs, of power series. through and new concepts arising from these subtleties are prod- But things turned out not ucts of the nineteenth century. The nineteenth century is to be so simple, and what hap- the ideas, also the source of the basics of complex analysis, from the pened next is the subject of Jer- Cauchy–Riemann equations to Riemannian surfaces and emy Gray’s thorough, detailed, results, and beyond. If you want to see how this all came about, you and fascinating book.1 Gray’s 29 want to read The Real and the Complex by Jeremy Gray. chapters take us on quite a ride difficulties In the eighteenth century, most mathematicians acted through the ideas, results, and as though symbolism and its heuristic power gave com- difficulties of the major figures plete insight into the general structure of mathematical of nineteenth-century analysis. As the century began, ideas. They moved back and forth between the real and analysts may have expected to succeed by continuing the complex, the finite and the infinite. The Bernoul- the calculational approach. But as they turned to Fourier lis, Taylor, Maclaurin, d’Alembert, Euler, Lagrange, and series, elliptic integrals, foundations of calculus, and Laplace studied functions by means of their power-series complex functions, earlier approaches did not suffice, representations, solved both ordinary and partial dif- and eighteenth-century problems and methods gave rise ferential equations, and developed methods of integra- to nineteenth-century theories and standards of proof. We tion. Integrals were defined as the inverse of derivatives. will hit some high points, hoping that readers will turn General statements about functions were accepted even if to the full text to experience the wealth and originality there were counterexamples, as long as the “exceptions” of the reasoning and ideas of the mathematicians of the nineteenth century. Judith V. Grabiner is the Flora Sanborn Pitzer Professor of Math- 1Gray intends this book as the second in a four-part series on ematics Emerita at Pitzer College, Claremont, California. Her email nineteenth-century mathematics, all arising from courses he has address is [email protected] . taught to advanced undergraduates. The first, Worlds Out of For permission to reprint this article, please contact: Nothing: A Course in the History of Geometry (Springer, 2011) [email protected]. has appeared; the last two are to cover differential equations and DOI: http://dx.doi.org/10.1090/noti1601 algebra.

1322 Notices of the AMS Volume 64, Number 11 Book Review

the book, Gray emphasizes how the real and the complex interacted in unexpected and Gray unpredictable ways. emphasizes In the 1820s, Cauchy began to treat real-variable calculus how the with a new level of rigor. He discredited Lagrange’s foun- real and dations by exhibiting a func- tion that did not equal its the complex Taylor series. He insisted that formulas hold “only under interacted in certain conditions and for cer- unexpected tain values of the quantities they contain.” He made the and Newtonian concept of limit precise and reasoned with unpredictable epsilons (borrowing many inequality techniques from ways. Lagrange) in key proofs about derivatives and in his system- atic treatment of the convergence of series. Instead of taking the existence of the integral for granted, Cauchy defined the definite integral as the limit of sums and gave a proof, assuming the function to be continuous (and, im- plicitly, uniformly continuous), that the limit existed. All this began the rigorization of the calculus, though much remained to be done. For instance, Cauchy thought he had proved that an infinite series of continuous functions was itself continuous. Abel found an “exception” (a Fourier series), but exactly what went wrong was not worked out Augustin-Louis Cauchy (1789–1857) began to treat real- until much later. variable calculus with a new level of rigor. The Real and the Complex lets readers follow the way that Cauchy moved in various directions in complex analysis as well. 1.5

Joseph Fourier, in his Analytical Theory of Heat of 1822, 1 operated like an eighteenth-century mathematician when he modeled how heat flows with a differential equation: 0.5 its solution was an infinite series of sine and cosine func- tions. Fourier said that all functions could be written as 0 2 4 6 8 10 12 such a series and said also that such a series converges x everywhere to the function it represents. Much later im- –0.5 portant work in analysis arose from figuring out when his statements are true and when they are not. –1 Beginning around 1790, Adrien-Marie Legendre studied elliptic integrals, another topic in real analysis with roots –1.5 in applied mathematics. He might have wanted to evalu- ate them exactly, but found that he could not, though he Figure 1. This graph was generated by the computer successfully reduced them to three canonical forms. Later, algebra system Maple and shows the first hundred Abel and Jacobi, each inverting Legendre-style elliptic terms of Abel’s function sin x – (1/2) sin 2x + (1/3) integrals by treating the upper endpoints as independent sin 3x –… . Gray remarks on “how unhappy Maple complex variables, discovered elliptic functions. The early gets at the points where the Abel function will be theory of elliptic functions remained formal; only later discontinuous.” This is one of a number of such were they brought into a general theory of complex func- instructive Maple graphs of functions with unusual tions, notably by Joseph Liouville’s theorem on doubly pe- properties in Gray’s book. riodic functions and Cauchy’s proof of it. So complex anal- ysis eventually made sense out of elliptic functions and elliptic integrals. In this episode, as in others throughout

December 2017 Notices of the AMS 1323 Book Review

Cauchy also recognized that complex functions could After a chapter on potential functions and Green’s be a separate object of study. Gray lets us follow the way theorem, Gray turns to the work of Dirichlet, who brought Cauchy moved in various directions, finding the Cauchy a new standard of rigor to much earlier work, in particular integral theorem early in the century, though not yet real- about the convergence of Fourier series. Rather than as- izing its fundamental importance. But by the 1840s and sume that every function has a convergent Fourier series 1850s, after many fits and starts, Cauchy had worked that equals it, Dirichlet asked under what conditions this was true. He found integral formulas for partial sums of the series and conditions for which they do converge. Gray illustrates Dirichlet’s reputation for rigor by quoting a let- ter from Jacobi to Alexander von Humboldt: “If Gauss says he has proved something, it seems very probable to me; if Cauchy says so, it is about as likely as not; if Dirichlet says so, it is certain.” Riemann, especially by emphasizing a conceptual rather than computational approach, transformed the theory of real and complex functions. As Gray puts it, “Riemann of- fered an approach to the study of functions that did not depend on any particular expression for it, and accord- ingly played down the role of long complicated manipu- lation of such expressions.” His introduction of trigono- metric series, even if the coefficients are not coefficients of a Fourier series, made the study of nondifferentiable functions possible. Defining analytic functions indepen- dently of the particular analytic expression for them by using the Cauchy–Riemann equations let Riemann relate them to harmonic functions. Riemann outlined a proof that two simply connected planar regions can be mapped conformally onto one another by complex analytic maps, now known as the Riemann mapping theorem. He showed how the geometry of complex curves illuminated the study of elliptic functions. And he defined the Riemann integral and enlarged the set of functions that can be integrated. Weierstrass had a different, more algebraic, approach Bernhard Riemann (1826–1866) emphasized a to complex function theory, choosing to emphasize the conceptual rather than computational approach and representation of complex analytic functions by conver- transformed the theory of real and complex functions. gent power series rather than the equivalent property of complex differentiability.3 The power-series approach let out many key properties of complex functions, including Weierstrass develop methods that apply to functions of what are now called the Cauchy–Riemann equations and one or several variables. He treated hyperelliptic and Abe- their relationship to differentiability, contour integrals, lian functions; made the concept of analytic continuation, Taylor-series expansions of complex functions, poles, and previously known to Riemann, central to his own theory; Laurent series. and distinguished between poles and essential singu- At this point in the narrative, Gray has demonstrated larities. His lectures at the University of Berlin widely dis- that much had changed in analysis. He reflects on what seminated his conclusions about continuity, convergence, these changes teach us in a brilliant review chapter, his uniform convergence, and term-by-term differentiability Chapter 12. These reflections enable readers to share in for both real and complex functions. In particular, he the experiences of leading mathematicians struggling with showed that there are continuous real functions that are problems and choices, occasional false trails, as well as nowhere differentiable. unexpected successes, never sure about where their ap- Gray next provides a chapter describing nineteenth- proaches would lead. Gray wants to make absolutely clear century work on uniform convergence, culminating in that this is what life is like when you do mathematics. If Weierstrass’s definitive and detailed account. Then, in you read nothing else in Gray’s book, do read this chapter,2 later chapters, Gray discusses topics for which some which makes this point using historical examples as well questions are answered in the nineteenth century while as I have ever seen it done. Similarly, the concluding sec- related questions are answered only in the twentieth. tions of many of his detail-packed chapters successfully Dirichlet and Riemann had called attention to functions convey the essence of what the chapter covered. that were not integrable or were defined by series that do not converge uniformly or that exhibit unusual relation-

2He names this chapter “Revision,” using the British term for 3Recall that Lagrange had tried to base his theory of real functions review. Similarly, he describes the book’s appendix reviewing on a detailed study of their real power series in his Théorie des potential theory as “a revision of the theory.” fonctions analytiques of 1797.

1324 Notices of the AMS Volume 64, Number 11 Book Review ships between their continuity and differentiability. Heine, Cantor, Hankel, Du Bois-Reymond, H. J. S. Smith, Darboux, and, especially, Weierstrass treated a variety of examples. No longer could it be assumed that functions are continu- ous and differentiable, except perhaps at isolated points, or that differentiable functions had to have integrable derivatives, and Bolzano, Riemann, Weierstrass, Hankel, Schwarz, and Dini began to consider functions in general and to ask which functions have the important properties of integrability and continuity. Gray describes how Weier- strass related implicit functions to holomorphic functions and how Dini made clear the distinction between the real and complex cases and also treated the several-variable case. Again, there were further developments in the twen- tieth century. Gray also addresses the completeness of the real numbers and the constructions of the reals from the rationals given by Weierstrass, Dedekind, Heine, and Cantor. Weierstrass realized that the Riemann integral was not sufficient, though he did not know how to make it so, and Gray later shows how such considerations led to Lebesgue’s theory of integration. Gray’s last three chapters explicitly address how prob- lems posed by nineteenth-century approaches moved toward the more modern and abstract ideas characteristic of the twentieth century, a process that resembles the way analysis had become both deeper and broader between the eighteenth century and the nineteenth. One chap- ter describes Lebesgue’s theory of integration. Another treats Cantor’s set theory and the foundations of mathemat- ics. The final chapter is about Karl Weierstrass (1815–1897) took a more algebraic Gray shows, topology. The reader may be approach to complex function theory than did rather eager for more forays into the Riemann. His lectures at the University of Berlin twentieth century, but Gray is widely disseminated much of his work. than tells, true to his goal to let us see gives credit to historians whose research helped inform nineteenth-century analysis in his own. His conclusions about controversial questions are providing a its own terms and not as a pre- generally well supported and judicious. He shows, rather great deal lude to the more abstract ideas than tells, as much as he can, providing a great deal of of the modern period. And he close-up mathematical detail. The writing style is clear and concludes with the statement unpretentious, as well as marked by immense learning. of close-up that “the mathematical analy- There are many ways one could write the history of sis of the last century also mathematical nineteenth-century analysis. Gray emphasizes the internal needs its historians.” Would development of the mathematics, though he does include that they do as good a job for detail. some biographical information about the lives of the the twentieth century as Gray mathematicians and their mutual relationships, especially has done for the nineteenth. As this brief overview may when he deems it crucial to explain how the mathematics indicate, Gray’s book is not light reading. But it is a very developed. Other historians might have said much more good book indeed. Any mathematician who would like to about how nineteenth-century society and culture affected know how analysis developed should find it interesting. mathematicians’ careers, choice of problems, the teaching Those who teach analysis can gain much insight from of mathematics, and the needs of the sciences, or might seeing the blind alleys as well as the great ideas, since have discussed in general the nineteenth-century profes- watching how the mathematics unfolded helps one ap- sionalization of mathematics, increase in specialization, preciate what can be really hard and what types of con- and journals and societies dedicated solely to mathemat- fusion can occur even to the most able student. Seeing ics. It would have been worth mentioning the work of Sofya past contrasting approaches to the same topic, like those Kovalevskaya and noting the institutional barriers that of Riemann and Weierstrass on complex functions, both prevented talented women from entering higher math- humanizes the subject and enhances understanding. And ematics. Of course both approaches mentioned here are Gray really knows his material, both the original sources valuable, and Gray’s choice is designed for his intended and the best of the current literature. He scrupulously audience.

December 2017 Notices of the AMS 1325 Book Review

Both my paperback review copy and the correspond- ing eBook version have more than the typical number of typographical errors. For instance, Cauchy’s function that ABOUT THE REVIEWER –1/x does not converge to its Taylor series is not e ; the ex- Judith V. Grabiner, author of The 2 ponent should be –1/x . But most are simply punctuation Origins of Cauchy’s Rigorous Cal- marks or words that are left out—in one case the word culus and “The role of mathematics “not”—and a second printing could correct these; a careful in liberal arts education” (in M. Mat- reader can usually navigate around them. Other matters thews, ed., International Handbook worth noting: Cauchy did not, in defining the definite of Research in History, Philosophy integral in terms of sums, restrict the mode of division and Science Teaching), received the of the interval of integration to equal subdivisions. And, Judith V. Grabiner Beckenbach Prize from the Math- especially since Gray makes a point of recommending ematical Association of America for accessible and instructive English-language sources, the her book A Historian Looks Back: The Bradley–Sandifer translation of Cauchy’s Cours d’analyse Calculus As Algebra and Selected ought to be added to the 12-page multilingual and com- Writings. prehensive bibliography. I strongly recommend The Real and the Complex to readers of the Notices. As a result of Gray’s vision and scholarship, any mathematically informed reader can now watch the giants of the nineteenth century, most notably Legendre, Fourier, Cauchy, Gauss, Dirichlet, Riemann, Wei- erstrass, Dedekind, and Cantor, developing much of the mathematics we use today—not­ flying straight as an arrow to their target, not possessing all our modern concepts, but doing the best they could so that, as Gray says, they “found their ways, imperfectly, from what they knew to what they wanted to know.”

ACKNOWLEDGMENT. I dedicate this review to the memory of Dr. Uta C. Merzbach (1933–2017), inspir- ing mentor, valued colleague, and beloved friend. Her long-anticipated study of Dirichlet, with the editorial assistance of Judy Green and Jeanne LaDuke, is forth- coming from Birkhäuser.

Image Credits Illustration of Augustin Cauchy courtesy of Library of Congress Prints and Photographs Division. From an illustration in: Das Neunzehnte Jahrhundert in Bildnissen / Karl Werckmeister, ed. Berlin : Kunstverlag der photographische Gesellschaft, 1901, vol. V, no. 581 Figure 1 by Jeremy Gray. Photo of Bernhard Riemann courtesy of Smithsonian Libraries Scientific Identity website. Photo of Karl Weierstrass courtesy of Smithsonian Libraries Sci- entific Identity website. Photo of Judith V. Grabiner courtesy of Pitzer College.

1326 Notices of the AMS Volume 64, Number 11 BOOKSHELF New and Noteworthy Titles on Our Bookshelf December 2017

One Two Three…Infinity: Facts The Universal Computer: The and Speculations of Science, by Road from Leibniz to Turing, by George Gamow (Dover, 1988) Martin Davis (AK Peters/CRC Press, This month’s Notices carries 2011) an article (see page 1275) about Next year, Martin Davis will George Gamow’s liquid model for turn 90 years old. A legendary the atomic nucleus, which has figure in mathematical logic, he is arisen in recent research and ap- best known for resolving Hilbert’s plications. Born in Odessa, Ukraine, Tenth Problem, together with Yuri in 1904, Gamow was a theoretical Matiyasevich, Hilary Putnam, and physicist and cosmologist with Julia Robinson. Davis also did wide interests ranging from radio- some of the world’s first computer active decay, to “big bang” cosmology, to genetics. He may programming, working in the 1950s on the ORDVAC be best known among mathematicians for his now-classic computer, which had a central memory consisting of 40 book highlighted here, One Two Three…Infinity. Originally vacuum tubes. A master expositor, Davis received the AMS published in 1947, the book has charmed readers for Steele Prize for Exposition as well as the Chauvenet and decades by presenting an appealing set of topics from Ford Prizes of the Mathematical Association of America. mathematics and science in simple, concrete terms that His popular book The Universal Computer originally ap- zoom in on the essential ideas. The book has a leisurely peared to wide acclaim in 2000 and was re-issued in 2012, feel that makes the reader feel like a participant in a tour in a special edition to honor the centenary of Alan Turing. led by a welcoming, enthusiastic, and highly knowledge- The tale of how computers developed has been told many able guide. The opening chapter discusses numbers and times and in many forms—and often with heavy empha- counting and even explains the notion of transfinite sis on the computer as an engineering feat. Davis takes numbers. Subsequent chapters discuss space, time, a completely different tack by tracing the origin of com- and Einstein’s theories of relativity, and the micro- and puters in developments in logic, starting with the ideas macro-cosmos. The book includes over 100 pen-and-ink of Leibniz, who, as Davis puts it, “dreamed of machines illustrations drawn by Gamow himself, which illustrate capable of carrying out calculations, freeing the mind for the ideas with humor and clarity and give a window into creative thought.” After an opening chapter on “Leibniz’s the mind of a man in love with ideas. Also known for his Dream,” Davis discusses the lives and work of Boole, Frege, series of popular science books featuring the inquisitive Cantor, Hilbert, Gödel, and Turing. The book closes with Mr. Tompkins, Gamow received the UNESCO Kalinga Prize a discussion of the first computers that were built as for the popularization of science in 1956. well as a look towards the future. “When a distinguished expert offers a popular exposition of his subject, we greet the effort with keen anticipation,” wrote Brian Blank in a review that appeared in the May 2001 issue of the Notices. “That is all the more true when the writer is as skilled as Martin Davis. It is a pleasure to report that in this case our anticipation is richly rewarded.”

Suggestions for the BookShelf can be sent to notices-booklist@ ams.org. We try to feature items of broad interest. Appearance of a book in the Notices BookShelf does not represent an endorsement by the Notices or by the AMS. For more, visit the AMS Reviews webpage www.ams.org/news/math-in-the-media/reviews.

December 2017 Notices of the AMS 1327 A MERICAN M ATHEMATICAL S OCIETY

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Contact the AMS Development Offi ce by phone: 401-455-4111 or email: [email protected] LETTERS TO THE EDITOR

Article about “Wide Influence” of L. E. Dickson The full space the article introduced in the Chinese name of can be considered an honest (though un- I read, and am fascinated by, the article on “The 楊克純 orthodox) attempt to highlight his family name, but it is ‘Wide Influence’ of Leonard Eugene Dickson,” by Della really unnecessary, as the correct form is to put his family Dumbaugh and Amy Shell-Gellasch in the August 2017 name first (before the given name), which is the standard Notices. It describes Dickson’s wide influence as reflected Chinese appellation—no spacing inserted. I would also like in three of his students graduating in the same year: A. to see Hua Luogeng's Chinese name ( ) used as well. Adrian Albert, Mina Rees, and Ko-Chuen Yang. 华罗庚 I thank the authors for writing such an inspiring article, All three are fascinating subjects by themselves. Albert which I have thoroughly enjoyed. is a well-known, notable algebraist of the Chicago school —Stanley Liu and an administrator of the highest order. Rees is the East Setauket, New York subject of a biography written by Shell-Gellasch. Yang and his student Hua Luogeng were fine contributors to (Received July 27, 2017) the development of the Chinese school of modern math- ematics. I am particularly interested in Yang, because he Updating Math Genealogy taught at Southwest Associated Universities, which had a seven-year tenure during the Sino-Japanse war and which In our article on “The ‘Wide Influence’ of Leonard Dickson” was attended by both my parents as well as others, such as in the Notices (August 2017), Amy Shell-Gellasch and I Yang’s son, the physics Nobel Laureate Chen-Ning (Frank) noted that Dickson had 67 PhD students in his forty-year Yang. (By the way, Southwest Associated Universities had career at the University of Chicago. Peter May inquired three presidents, whose disparate personalities were often about this number, since the Mathematical Genealogy the subject of good-natured humor and banter.) Project listed only 53 students for Dickson. Following the Admittedly, I am an admirer also of Leonard Eugene guidelines for submitting new students to the Mathemati- Dickson, whose seven-volume Mathematical Papers still cal Genealogy website, I added 15 names to Dickson’s list sit on the shelves of my personal library, which I have for and removed one name, Marion Stark, who was not his some time vowed to downsize, along with the rest of my PhD student. I want to extend a generous thanks to Peter hoard in the house. May for calling this discrepancy to our attention and to I applaud the authors and the Notices for including the Mathematical Genealogy Project for their excellent the Chinese name (in Unicode) of Ko-Chuen Yang, for service to the mathematical community. I encourage other disambiguation, because Chinese mathematicians often members of our community to submit updates to this published with names in Anglicized forms. I know six or valuable database. seven Chinese transcription systems, including the now- —Della Dumbaugh standard Hanyu Pinyin and others that preceded it (such as University of Richmond [email protected] Wade-Giles and Gwoyeu Romatzyh)—Shiing-Shen Chern’s name belongs to the latter system. People used whatever official system was then in force in official dictionaries for (Received August 25, 2017) these transcriptions. (The doubling of “i” in “Shiing” de- Peripatetic Accent Marks notes third tone; the seemingly extraneous “r” in “Chern” denotes second tone—these were concocted before the I would like to reassure the readers that the accent marks introduction of diacritical marks in computerized type- in my name are firmly rooted on the letters "a" and they setting and typography.) Besides, in the old Wade-Giles have no business wandering over to the "o" as in the Sep- system, there were mispronunciations, or dialectical influ- tember 2017 Notices (p. 922 column 2 and p. 923). ences, so that the system was many-to-one for the same —János Kollár Chinese characters. So, I believe supplying the Chinese Department of Mathematics names of Chinese scientists helps a lot to disambiguate. Princeton University [email protected]

*We invite readers to submit letters to the editor to notices- (Received September 11, 2017) [email protected] and post commentary on the Notices webpage www.ams.org/journals/notices. EDITOR’S NOTE. Yes, sorry, it is a shame to spoil such For permission to reprint this article, please contact: nice symmetry. [email protected].

December 2017 Notices of the AMS 1329 THE BACK PAGE

“All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But they also live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to another. They’re adults in the crystalline world, infants in the real one.”—Sylvain Cappell

Concept and artwork by Anna Pun and Richard Stanley.

Robert McCann reports that his young son Ian was looking for words beginning with “lem.” Robert responded, “Lemon, lemma, and lemniscate, although I don't know what that is.” Ian asked, “Daddy, are there mathematicians better than you who know what a lemniscate is?”

What crazy things happen to you? Readers are invited to submit original short amusing stories, math jokes, cartoons, and other material to: [email protected].

1330 Notices of the AMS Volume 64, Number 11 NEWS

Mathematics Opportunities

Listings for upcoming math opportunities to appear in Notices may be submitted to [email protected] .

massmediafellow and through the AMS Washington Of- AMS–AAAS Mass Media fice, 1527 Eighteenth Street, NW, Washington, DC 20036; Summer Fellowships telephone 202-588-1100; email [email protected]. The deadline for applications is January 15, 2018. The American Mathematical Society provides support each year for a graduate student in the mathematical sci- —AMS Washington Office ences to participate in the American Association for the Advancement of Science (AAAS) Mass Media Science and Engineering Fellows Program. This summer fellowship *NSF Program in program pairs graduate students with major media outlets nationwide, where they will research, write, and report Computational and on science news and use their skills to bring technical subjects to the general public. Data-Enabled Science and The principal goal of the program is to increase the Engineering in Mathematical public’s understanding of science and technology by strengthening the connection between scientists and and Statistical Sciences journalists to improve coverage of science-related issues The Program in Computational and Data-Enabled Science in the media. Past AMS-sponsored fellows have held posi- and Engineering in Mathematical and Statistical Sciences tions at National Public Radio, Scientific American, Voice (CDS&E-MSS) of the National Science Foundation accepts of America, The Oregonian, NOVA, the Chicago Tribune, proposals that confront and embrace the host of math- and the Milwaukee Journal Sentinel. ematical and statistical challenges presented to the scien- Fellows receive a weekly stipend of US$500, plus travel tific and engineering communities by the ever-expanding expenses, to work for ten weeks during the summer as role of computational modeling and simulation on the reporters, researchers, and production assistants in news- one hand and the explosion in production of digital and rooms across the country. They observe and participate in observational data on the other. The CDS&E-MSS program the process by which events and ideas become news, im- supports fundamental research in mathematics and prove their ability to communicate about complex techni- statistics whose primary emphasis is on meeting these cal subjects in a manner understandable to the public, and challenges. The window for submission is November increase their understanding of editorial decision making 25–December 11, 2017. See www.nsf.gov/funding/ and of how information is effectively disseminated. Each pgm_summ.jsp?pims_id=504687&org=DMS&from=home. fellow attends an orientation and evaluation session in Washington, DC, and begins the internship in mid-June. —NSF announcement Fellows submit interim and final reports to AAAS. A wrap- up session is held at the end of the summer. Mathematical sciences faculty are urged to make their graduate students aware of this program. Further information about the fellowship program and applica- tion procedures is available online at www.aaas.org/ programs/education/MassMedia; or applicants may *The most up-to-date listing of NSF funding opportunities from contact Rebekah Corlew, Program Director, AAAS Mass the Division of Mathematical Sciences can be found online at: Media Science and Engineering Fellows Program, 1200 New www.nsf.gov/dms and for the Directorate of Education and York Avenue, NW, Washington, DC 20005; email rcorlew@ Human Resources at www.nsf.gov/dir/index.jsp?org=ehr. aaas.org . Further information is also available at www. To receive periodic updates, subscribe to the DMSNEWS listserv by ams.org/programs/ams-fellowships/media-fellow/ following the directions at www.nsf.gov/mps/dms/about.jsp.

December 2017 Notices of the AMS 1331 Mathematics Opportunities NEWS National Defense Science MSRI Summer Research for and Engineering Graduate Women in Mathematics Fellowships MSRI invites applications for participation in the Summer Research for Women in Mathematics program (June 11, To help increase the number of US citizens trained in 2018–August 3, 2018). disciplines of military importance in science and engi- Groups of two to six women with partial results on an neering, the Department of Defense awards National established project may submit an application to the Defense Science and Engineering Graduate Fellowships program. Each member of the group must have a PhD in to individuals who have demonstrated ability and special mathematics or advanced graduate standing. aptitude for advanced training in science and engineering. The deadline for application will be February 1, 2018. The fellowships are awarded for a period of up to four Financial Support for travel and local expenses is avail- years for study and research leading to doctoral degrees able. For more information, please see www.msri.org/ in any of fifteen scientific disciplines. Application forms programs/329. are available online at www.ndsegfellowships.org/ap- plication and are due December 31, 2017. —From an MSRI announcement

—From an NDSEG announcement

CRM Intensive Research Programs The Centre de Recerca Matemàtica (CRM) will hold two Intensive Research Programs in the spring of 2018. An advanced course in “Recent Progress in Mathematical Biol- ogy” will be held April 3–6, 2018, with a conference to be held June 4–8, 2018. A course in “Discrete, Combinatorial, and Computational Geometry” will be held from April 16–June 8, 2018. See www.crm.cat/en/Host/SciEvents/ IRP/Pages/CurrentIRP.aspx.

—From a CRM announcement

STaR Fellowship Program MATHEMATICAL SCIENCE The Service, Teaching, and Research (STaR) Program of OPPORTUNITIES FROM THE AMS the Association of Mathematics Teacher Educators (AMTE) supports the development of early-career mathematics educators, including their induction into the profes- The AMS Online Opportunities Page provides sional community of university-based teacher educators another avenue for the math community to and researchers in mathematics education. Senior and Announce and Browse: mid-career mathematics education faculty organize and facilitate STaR events, serving as mentors to Fellows. Ap- plications are due December 1, 2017; see www.amte.net/ • Calls for fellowship appointments star/apply. • Prize and award nominations —From an AMTE announcement • Grant applications

• Meeting and workshop proposals

search now! www.ams.org/opportunities

1332 Notices of the Ams VoLUme 64, NUmber 11 Unique Fellowship Opportunity An Invitation to Apply for the AMS-AAAS Congressional Fellowship

The American Mathematical Society (AMS), in conjunction with the American Association for the Advancement of Science (AAAS), will sponsor a Congressional Fellow from September 2018 through August 2019. The Fellow will spend the year working on the sta of a Member of Congress or a congressional committee, working as a special legislative assistant in legislative and policy areas requiring scienti c and technical input. The program includes an orientation on congressional and executive branch operations, and a year- long seminar series on issues involving science, technology and public policy. The Fellowship is an extraordinary rite The Fellowship is designed to provide a unique public policy learning of“ passage to a new world of discovery. Not experience, to demonstrate the value of science-government interaction, and only will you learn about policymaking and to bring a technical background and external perspective to the decision- government a airs, you’ll learn more about ” making process in the Congress. yourself and how you can impact the world. — Carla Cotwright-Williams Prospective Fellows must demonstrate expertise in some area of the AMS Congressional Fellow 2012–2013 mathematical sciences; have a good scienti c and technical background; be cognizant of and demonstrate sensitivity toward political and social issues; It was thrilling to be intricately involved and, most importantly, have a strong interest and some experience in applying in“ the drafting of legislation. My mathematical personal knowledge toward the solution of societal problems. and procedural training enabled me to focus on the core of an issue and craft legislative Applications are invited from individuals in the mathematical sciences. proposals that may become part of policy Applicants must have a PhD or an equivalent doctoral-level degree in aimed at improving society.” mathematics by the application deadline (February 15, 2018). Applicants must be US citizens. Federal employees are not eligible. — Anthony Macula, AMS Congressional Fellow 2015–2016 An AMS Fellowship Committee will select the AMS Congressional Fellow. The Fellowship stipend is US$79,720 for the fellowship period, with allowances for The AMS Congressional Fellowship is an relocation and professional travel and a contribution toward health insurance. incredible“ opportunity to directly engage with critical issues that impact our lives, research, To apply online, please go to bit.ly/AMSCongressionalFellowship. Candidates and careers. You will gain valuable legislative must submit a statement expressing interest and quali cations for the AMS experience, build powerful networks, and Congressional Fellowship, as well as current curriculum vitae. Candidates must come away with expertise that can support also arrange for three letters of recommendation to accompany the application. continued civic engagement.” — Catherine Paolucci, DEADLINE FOR RECEIPT OF APPLICATIONS: February 15, 2018. AMS Congressional Fellow 2016–2017 For more information, please contact [email protected] or call the AMS Washington Office at 202-588-1100.

AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE NEWS

Mathematics People

smooth boundaries. Akito Suzuki of Shinshu University Tikhomirov Awarded CMS was recognized for a spectral and scattering theoretic Doctoral Prize proof of the weak limit theorem for quantum walks. Shuya Chiba of Kumamoto University was honored for Konstantin Tikhomirov of his work on 2-factors containing perfect matchings in bi- Princeton University has been partite graphs and directed 2-factors in digraphs. Junya awarded the 2017 Doctoral Prize of Nishiguchi of Kyoto University was recognized for his the Canadian Mathematical Society work entitled “How Should We Understand the Time-Delay (CMS) for his “outstanding contribu- Structure in Dynamics?” Yuto Miyatake of Nagoya Uni- tions to several open problems in versity received the prize for his work on finite difference probability theory, convex geometry, schemes preserving multiple invariants for evolutionary functional analysis, and random ma- partial differential equations. trix theory.” According to the prize citation, in his doctoral studies, —From an MSJ announcement Konstantin Tikhomirov investigated “a series Tikhomirov of open problems in diverse areas of mathematics. He has written at least 2017 Davidson Fellows Chosen ten papers related to asymptotic geometric analysis, ran- dom matrices, probability theory, and convex geometry. Two high school students whose projects involved the mathematical In particular, he worked on the problem of estimating the sciences have been named 2017 Da- distance between an n-dimensional polytope with a fixed vidson Fellows. Arjun Ramani, eigh- number of vertices and the Euclidian ball. In this case he teen, of West Lafayette, Indiana, was solved the exact dependence between the dimension and awarded a US$25,000 scholarship the number of vertices. for his project “Fast Sampling of Sto- “Tikhomirov also considered problems like understand- chastic Kronecker Graphs.” Ramani ing better the limit of the smallest singular value of some also won third place in this year’s families of random matrices, as well as when the convex Regeneron Science talent search. He hull of a random walk includes the origin. The impact of Arjun Ramani tells the Notices: “I recently created a his work will have many ramifications for its innovation rap song on GarageBand that I have and its ability to be applied in other situations.” posted online on Soundcloud. I have Tikhomirov received his PhD from the University of played competitive tennis since the Alberta in 2016. The Doctoral Prize is awarded annually age of nine. I became interested in to a doctoral student from a Canadian university who has statistics because of my love of NBA demonstrated exceptional performance in mathematical basketball at a young age.” research. Felix Wang, eighteen, of New- —From a CMS announcement ton, Massachusetts, was awarded a US$25,000 scholarship for his proj- ect “Functional Equations in Com- plex Analysis and Number Theory.” Prizes of the Mathematical Felix Wang The Davidson Fellows program, Society of Japan a project of the Davidson Institute for Talent Development, awards scholarships to students The Mathematical Society of Japan (MSJ) has awarded its eighteen years of age or younger who have created signifi- Prizes for Excellent Applied Mathematicians for 2017. To- moya Memmochi of the University of was honored for the study of L∞-error estimates for the finite element approximation of parabolic problems on domains with

1334 Notices of the AMS Volume 64, Number 11 Mathematics People NEWS cant projects that have the potential to benefit society in •• William Thomason, computer and computational sci- the fields of science, technology, mathematics, literature, ences, Cornell University, ONR music, and philosophy. •• Brian Thompson, computer and computational sci- ences, ARO —From a Davidson Fellows announcement •• Marvin Zhang, computer and computational sci- ences, University of California Berkeley, ARO NDSEG Fellowships Awarded —From a DoD announcement Twenty-one young scholars whose work involves the math- ematical sciences have been awarded National Defense Sci- CME Group–MSRI Prize ence and Engineering Graduate (NDSEG) Fellowships by the Department of Defense (DoD) for 2017. The fellowships Awarded are sponsored by the United States Army, Navy, and Air Force. As a means of increasing the number of US citizens Robert B. Wilson of the Graduate trained in disciplines of military importance in science School of Business has been awarded the CME Group–MSRI and engineering, DoD awards fellowships to individuals Prize in Innovative Quantitative Applications. His research and teaching focus on market design, pricing, negotiation, who have demonstrated ability and special aptitude for and related topics concerning industrial organization and advanced training in science and engineering. information economics. He is an expert on game theory Following are the names of the fellows, their research and its applications. fields, their institutions, and the offices that awarded the fellowships: —From a CME–MSRI announcement •• Dallas Albritton, mathematics, University of Minne- sota-Twin Cities, Army Research Office (ARO) •• Brandon Bohrer, computer and computational sci- ences, Carnegie Mellon University, Air Force Research Malliaris and Shelah Awarded Laboratory (AFRL) •• Steven Brill, computer and computational sciences, Hausdorff Medal Stanford University, AFRL Maryanthe Malliaris of the University of Chicago and •• Victoria Chayes, mathematics, Rutgers University, of the Hebrew University of Jerusalem AFRL and Rutgers University have been awarded the third •• Caleb Chuck, computer and computational sciences, Hausdorff Medal for their joint paper “Cofinality Spec- University of Texas at Austin, Office of Naval Research trum Theorems in , Set Theory, and General (ONR) Topology,” published in the Journal of the American •• Tanner Fiez, computer and computational sciences, Mathematical Society 29 (2016). The Hausdorff Medal University of Washington, ARO is awarded every two years by the European Set Theory •• Yakir Forman, mathematics, , AFRL Society (ESTS) for the most influential work in set theory •• Jesse Freeman, mathematics, Massachusetts Institute published in the preceding five years. of Technology, ONR •• Pranav Gokhale, computer and computational sci- —From an ESTS announcement ences, University of Chicago, AFRL •• Benjamin Gunby, mathematics, , ARO •• Tianshe He, mathematics, University of North Carolina Royal Society of at Chapel Hill, ONR Canada Fellows •• Siddhartha Jayanti, computer and computational sciences, Massachusetts Institute of Technology, ARO The Royal Society of Canada (RSC) has elected two •• Samuel McLaren, mathematics, University of Arizona, mathematical scientists to its 2017 class of new fellows AFRL in mathematical sciences. They are Yoshua Bengio of •• Matthew McMillan, mathematics, University of Cali- the University of Montreal and Robert Jerrard of the fornia Los Angeles, ARO University of Toronto. •• David Mildebrath, computer and computational sci- ences, , ARO —From an RSC announcement •• Brandon Shapiro, mathematics, Cornell University, Photo Credits AFRL •• Mackenzie Simper, mathematics, Stanford University, Photo of Konstantin Tikhomirov courtesy of Konstantin ONR Tikhomirov •• Samuel Stewart, mathematics, University of Minne- Photos of Arjun Ramani and Felix Wang courtesy of Davidson sota-Twin Cities, AFRL Institute for Talent Development.

December 2017 Notices of the AMS 1335 NEWS

Inside the AMS

The day will be structured to include and encourage net- Project NExT 2017 working and sharing of ideas among participants. There Fellows Chosen will be four sessions: (1) View from the top. What responsibilities, duties, and Six mathematicians have been selected as AMS Project expectations do deans, provosts, and other chief academic NExT Fellows for 2017. Their names and affiliations follow. officers have for their chairs? •• Edgar A. Bering, Temple University (2) Improving students’ experience. Possible topics •• Sara Clifton, University of Illinois at Urbana—Cham- include curriculum and research opportunities, student paign recruitment and diversity, program assessment, career •• Patrick Devlin, Yale University counseling, and also personnel issues such as faculty •• Brian Hwang, Cornell University development and incentives and the increasing numbers •• Pamela Beth Pyzza, Ohio Wesleyan University of non-tenure-track faculty. •• Amelia Tebbe, Indiana University (3) Outreach and communication: Building effective in- Project NExT (New Experiences in Teaching) is a ternal partnerships. Possible topics include collaborations professional development program for new and recent with other departments, working with university offices PhDs in the mathematical sciences (including pure and such as honors programs, government relations offices, applied mathematics, statistics, operations research, and career and internship offices, development office, and the mathematics education). It addresses all aspects of an dean and upper administration. academic career: improving the teaching and learning of (4) Outreach and communication: Building effective mathematics, engaging in research and scholarship, and external partnerships. Possible topics include collabora- participating in professional activities. It also provides the tions with local businesses, local school systems, and participants with a network of peers and mentors as they assume these responsibilities. The AMS provides funding other regional or national efforts. for a number of the Fellowships. The workshop registration fee of US$150 is in addi- tion to and separate from the Joint Meetings registra- —From an MAA announcement tion. Those interested in attending can register at bit. ly/2xtP45U by December 20, 2017. For further informa- tion, please contact the AMS Washington Office at 202- AMS Department 588-1100 or [email protected] . Chairs Workshop —Anita L. Benjamin AMS Washington Office The annual workshop for department chairs will be held a day before the start of the Joint Mathematics Meetings in San Diego, California, on Tuesday, January 9, 2018, from 8:00 am to 6:30 pm at the Marriott Marquis San Diego Free Grant-Writing Marina hotel. Workshop Offered Workshop leaders will be Malcolm Adams, former head, Department of Mathematics, University of Georgia; Krista The American Mathematical Society, in conjunction with Maxson, vice president of academic affairs, University of the National Science Foundation Directorate for Education Science and Arts of Oklahoma and former chair of the and Human Resources (NSF-EHR), is pleased to offer a free Department of Mathematical Sciences at Shawnee State workshop entitled “Writing a Competitive Grant Proposal University; Irina Mitrea, chair, Department of Mathematics, to NSF-EHR.” This grant-writing workshop will be held on Temple University; and Douglas Mupasiri, head, Depart- Monday, January 8, 2018, from 3:00 pm to 6:00 pm at the ment of Mathematics, University of Northern Iowa. Marriott Marquis San Diego Marina hotel. What makes a chair different from any other engaged Workshop Goals: faculty member in the department? This one-day work- •• To familiarize participants with current direction/ shop will examine the chair’s role in leading a department. priorities in EHR

1336 Notices of the AMS Volume 64, Number 11 Inside The AMS NEWS

•• To familiarize participants with key EHR education Find the research and development programs •• To consider common issues of competitive proposals data of interest •• To prepare participants to write a competitive proposal Topics covered will include discussion of key programs in EHR; the merit review process and merit review crite- to you! ria; discussion of scenarios—short passages drawn from www.ams.org/annual-survey proposals in the EHR portfolio designed to stimulate dis- cussion about strengths and weaknesses of a proposal; and opportunities to discuss possible proposal ideas with program officers. This free workshop is open to all interested participants by registering by December 20, 2017, at bit.ly/2fz1R0a. For further information, please contact the AMS Washing- ton Office at 202-588-1100 or [email protected]. New Doctoral Recipients Find out about new PhD graduates, —Anita L. Benjamin their employment plans, demographics, AMS Washington Office and starting salaries. Doctoral Degrees and Thesis Titles See a listing of PhD graduates, their thesis titles, and where they earned their degrees. Faculty Salaries See the recent academic salaries for full-time tenured and tenure-track Mathematical Sciences faculty. Recruitment & Hiring Get a comprehensive picture of the academic job market in the Mathematical Sciences. Twenty Years Ago in the Departmental Profile Notices Learn about the number of faculty in various categories, undergraduate and graduate course The Football Player and the Infinite Series enrollments, the number of graduate students, by Harold P. Boas and the number of masters and bachelors degrees awarded in the Mathematical Sciences. www.ams.org/notices/199711/boas.pdf

The star football player—soccer player in Ameri- can lingo—referred to in the article title played in the 1908 Olympics on 's silver-medal soccer team. But this player was also an out- standing mathematician. When he defended his thesis, "the air buzzed with anticipation as the football team crowded excitedly into the lecture hall." The title of the dissertation was Contribu- tions to the Theory of Dirichlet Series, and the candidate's name was . This article discusses Bohr's thesis work involving Dirichlet series and the Riemann zeta function. The article author, Harold P. Boas, an outstand- ing expositor and recipient of the Bergman Prize, Annual Survey became editor of the Notices three years after of the Mathematical Sciences this article was published. AMS • ASA • IMS • MAA • SIAM

December 2017 Notices of the AMS 1337 NEWS

For Your Information

Important: To streamline this year’s process for all Departments Coordinate Job involved, we ask that you notify T. Christine Stevens at Offer Deadlines the AMS ([email protected]) if and only if: (1) your department is not listed and you would like to For the past eighteen years, the American Mathematical be listed as part of the agreement; Society has led the effort to gain broad endorsement for or the following proposal: (2) your department is listed and you would like to with- That mathematics departments and institutes agree draw from the agreement and be removed from the list. not to require a response prior to a certain date (usually Please feel free to e-mail us with questions and con- around February 1 of a given year) to an offer of a post- . Thank you for consideration of the proposal. doctoral position that begins in the fall of that year. This proposal is linked to an agreement made by the —Catherine A. Roberts National Science Foundation (NSF) that the recipients of AMS Executive Director the NSF Mathematical Sciences Postdoctoral Fellowships T. Christine Stevens would be notified of their awards, at the latest, by the AMS Associate Executive Director end of January. This agreement ensures that our young colleagues entering the postdoctoral job market have as much in- The EDGE Program Turns 20 formation as possible about their options before making a decision. It also allows departmental hiring committees adequate time to review application files and make in- formed decisions. From our perspective, this agreement has worked well and has made the process more orderly. There have been very few negative comments. Last year, one hundred seventy-seven (177) mathematical sciences departments and four (4) mathematics institutes endorsed the agreement. Therefore we propose that mathematics departments again collectively enter into the same agreement for the upcoming cycle of recruiting, with the deadline set for Monday, January 29, 2018. The NSF’s Division of Math- ematical Sciences has already agreed that it will complete its review of applications by January 19, 2018, at the lat- est, and that all applicants will be notified electronically at that time. EDGE@20 The American Mathematical Society facilitated the pro- The EDGE Program (Enhancing Diversity in Graduate Edu- cess by sending an e-mail message to all doctoral-granting cation) will celebrate its twentieth anniversary in 2018. mathematics and applied mathematics departments and Since 1998, the program’s four-week summer session, mathematics institutes. The list of departments and in- which initially alternated between Bryn Mawr and Spelman stitutes endorsing this agreement was widely announced Colleges, has been held at colleges and universities around on the AMS website beginning November 1, 2017, and is the country. The current codirectors are Ami Radunskaya updated weekly until mid-January. of Pomona College and Raegan Higgins of Texas Tech We ask that you view this year’s formal agreement at University. With a mission to prepare a diverse group of www.ams.org/employment/postdoc-offers.html along women students for the academic and cultural demands with this year’s list of adhering departments. of graduate programs in the mathematical sciences, the

1338 Notices of the AMS Volume 64, Number 11 For Your Information NEWS A MERICAN M ATHEMATICAL S OCIETY program has hosted 241 students to date, 78 of whom have received PhDs. Many more have earned master’s de- grees, are currently pursuing advanced degrees, or have pursued careers in academia, government, industry, K–12 education, finance, and entrepreneurship. EDGE alumnae Give the have received NSF, Fulbright, Alliance, and Congressional Fellowships, chaired their departments, become deans, de- Gift of signed outreach programs, started collaborative research groups, and made countless positive changes in the world. In 2013, the nonprofit Sylvia Bozeman and Rhonda Hughes EDGE Foundation was established to ensure the financial future of the EDGE Program (www.edgefounda- AMS tion.net). Throughout 2018, a fundraising campaign, EDGE@20, as well as special sessions at the Joint Math- ematics Meetings in San Diego and celebrations around the Membership country, will mark the program’s twenty-year milestone. Visit EDGE at facebook@edge4women or at www.edge- forwomen.org . Surprise someone with the gift of connection to the mathematical community. Your —EDGE Program announcement purchase of an AMS Membership supports Photo Credit the AMS in its mission to further EDGE@20 photo by Maia Averett, Mills College. mathematical research, scholarship, professionalism, education, and awareness. It is a gift that keeps on giving.

There are three easy steps to make your gift membership purchase:

1) Visit www.ams.org/membership/ member-application-18.pdf to download the AMS Membership application.

2) Complete the application for the gift recipient.

3) Mail the completed application with payment. Include a simple letter that indicates the membership is a gift. Mail to:

American Mathematical Society Sales and Member Services Department 201 Charles Street Providence, RI 02904-2294 USA

If you have questions, please contact the Sales and Member Services Department at: [email protected] or 800.321.4267.

December 2017 Notices of the AMS 1339 A MERICAN M ATHEMATICAL S OCIETY WILL YOU BE ATTENDING the Joint Mathematics Meetings in San Diego?

Visit the AMS Membership Booth AMS Members, get your professional to learn more about the benefi ts portrait taken at the Membership of membership! Booth! The AMS is excited to announce a NEW The AMS Membership Department has benefi t available to all individual members, arranged for a photographer to take your FREE SHIPPING! In addition to receiving a professional portrait and have it emailed to discount on books purchased through the you in just a few minutes! You can upload this online bookstore and at meetings, members photo to your MathSciNet® Author Profi le are also entitled to receive free shipping on page, use it on your University website, submit their purchases. Join or renew your it as the professional photograph for your membership and receive a complimentary gift! book publication, or use it as your profi le picture in email and on social platforms. Availability: Thursday, January 11th 2018, 9:30 am– 12:30 pm and Friday, January 12th 2018, 2:30 pm–5:30 pm. Schedule your appointment at: amermathsoc.simplybook.me New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to www.ams.org/bookstore-email.

Algebra and Algebraic Orthogonal and Geometry Symplectic 푛-level Densities Tensor Products and A. M. Mason, , United Kingdom, and Regularity Properties N. C. Snaith, University of Bristol, of Cuntz Semigroups United Kingdom

Ramon Antoine, Universitat Contents: Introduction; Eigenvalue Autónoma de Barcelona, Spain, statistics of orthogonal matrices; Francesc Perera, Universitat Eigenvalue statistics of symplectic matrices; 퐿-functions; Zero Autónoma de Barcelona, Spain, statistics of elliptic curve 퐿-functions; Zero statistics of quadratic Dirichlet 퐿-functions; 푛-level densities with restricted support ; and Hannes Thiel, Universität Example calculations; Bibliography. Münster, Germany Memoirs of the American Mathematical Society, Volume 251, This item will also be of interest to those working in number Number 1194 theory. December 2017, 93 pages, Softcover, ISBN: 978-1-4704-2685-9, Contents: Introduction; Pre-completed Cuntz semigroups; 2010 Mathematics Subject Classification: 11M50; 15B52, 11M26, Completed Cuntz semigroups; Additional axioms; Structure of 11G05, 11M06, 15B10, Individual member US$47, List US$78, Cu-semigroups; Bimorphisms and tensor products; Cu-semirings Institutional member US$62, Order code MEMO/251/1194 and Cu-semimodules; Structure of Cu-semirings; Concluding remarks and open problems; Appendix A. Monoidal and enriched categories; Appendix B. Partially ordered monoids, groups and rings; Bibliography; Index of terms; Index of symbols. La Formule des Traces Memoirs of the American Mathematical Society, Volume 251, Locale Tordue Number 1199 December 2017, 191 pages, Softcover, ISBN: 978-1-4704-2797-9, Colette Mœglin, Institut de 2010 Mathematics Subject Classification: 06B35, 06F05, 15A69, Mathématiques de Jussieu-CNRS, 46L05; 06B30, 06F25, 13J25, 16W80, 16Y60, 18B35, 18D20, 19K14, France, and J.-L. Waldspurger, 46L06, 46M15, 54F05, Individual member US$47, List US$78, Institut de Mathématiques de Institutional member US$62, Order code MEMO/251/1199 Jussieu-CNRS, France

This item will also be of interest to those working in number theory. Contents: La formule des traces locale tordue ; La formule des traces locale tordue sous forme symétrique; Index des notations, par ordre alphabétique et par chapitre; Bibliography. Memoirs of the American Mathematical Society, Volume 251, Number 1198 December 2017, 180 pages, Softcover, ISBN: 978-1-4704-2771- 9, Individual member US$47, List US$78, Institutional member US$62, Order code MEMO/251/1198

December 2017 Notices of the AMS 1341 New Publications Offered by the AMS Rational Points on Analysis Varieties Bjorn Poonen, Massachusetts On Sudakov’s Type Institute of Technology, Decomposition of Cambridge, MA Transference Plans This book is motivated by the problem with Norm Costs of determining the set of rational points on a variety, but its true goal is to equip Stefano Bianchini, SISSA, readers with a broad range of tools essential for current research in algebraic Trieste, , and Sara Daneri, geometry and number theory. The book is unconventional in Friedrich-Alexander-Universität that it provides concise accounts of many topics instead of a Erlangen-Nürnberg, Germany comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included Contents: Introduction; General notations and definitions; are Brauer groups, faithfully flat descent, algebraic groups, Directed locally affine partitions on cone-Lipschitz foliations; torsors, étale and fppf cohomology, the Weil conjectures, and the Proof of Theorem 1.1; From C̃푘-fibrations to linearly ordered Brauer-Manin and descent obstructions. A final chapter applies all C̃푘-Lipschitz foliations; Proof of Theorems 1.2–1.6; Appendix A. these to study the arithmetic of surfaces. Minimality of equivalence relations; Chapter B. Notation; Chapter The down-to-earth explanations and the over 100 exercises make C. Index of definitions; Bibliography. the book suitable for use as a graduate-level textbook, but even Memoirs of the American Mathematical Society, Volume 251, experts will appreciate having a single source covering many Number 1197 aspects of geometry over an unrestricted ground field and December 2017, 112 pages, Softcover, ISBN: 978-1-4704-2766-5, containing some material that cannot be found elsewhere. 2010 Mathematics Subject Classification: 28A50, 49Q20, Individ- The origins of arithmetic (or Diophantine) geometry can be traced ual member US$47, List US$78, Institutional member US$62, back to antiquity, and it remains a lively and wide research Order code MEMO/251/1197 domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central Spaces subject: obstructions to existence of rational points. An Introduction to Real —, Max-Planck-Institute, Bonn Analysis It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a Tom L. Lindstrøm, University of student. Oslo, Norway

—Hendrik Lenstra, University Leiden Spaces is a modern introduction to real Understanding rational points on arbitrary algebraic varieties analysis at the advanced undergraduate is the ultimate challenge. We have conjectures but few results. level. It is forward-looking in the sense Poonen’s book, with its mixture of basic constructions and that it first and foremost aims to openings into current research, will attract new generations to the provide students with the concepts and techniques they need Queen of Mathematics. in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are —Jean-Louis Colliot-Thelene, Université Paris-Sud a solid understanding of calculus and linear algebra. Two A beautiful subject, handled by a master. introductory chapters will help students with the transition from —Joseph Silverman, computation-based calculus to theory-based analysis. This item will also be of interest to those working in number The main topics covered are metric spaces, spaces of continuous theory. functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although Contents: Fields; Varieties over arbitrary fields; Properties of some of the topics are more advanced than what is usually found morphisms; Faithfully flat descent; Algebraic groups; Étale in books of this level, care is taken to present the material in a way and fppf cohomology; The Weil conjecture; Cohomological that is suitable for the intended audience: concepts are carefully obstructions to rational points; Surfaces; Universes; Other kinds of introduced and motivated, and proofs are presented in full detail. fields; Properties under base extension; Bibliography; Index. Applications to differential equations and Fourier analysis are Graduate Studies in Mathematics, Volume 186 used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their December 2017, 337 pages, Hardcover, ISBN: 978-1-4704-3773-2, skills and understanding. The text has been tested in classes at the LC 2017022803, 2010 Mathematics Subject Classification: 14G05; over a number of years. 11G35, AMS members US$66.40, List US$83, Order code GSM/186 Contents: Introduction – Mainly to the students; Preliminaries: Proofs, sets, and functions; The foundation of calculus; Metric spaces; Spaces of continuous functions; Normed spaces and linear operators; Differential calculus in normed spaces; Measure and

1342 Notices of the AMS Volume 64, Number 11 New Publications Offered by the AMS integration; Constructing measures; Fourier series; Bibliography; Geometry and Topology Index. Pure and Applied Undergraduate Texts, Volume 29 January 2018, 369 pages, Hardcover, ISBN: 978-1-4704-4062-6, Bounded Cohomology LC 2017022199, 2010 Mathematics Subject Classification: 26-01, of Discrete Groups 28-01, 42-01, 46-01, 54E35, 26E15, AMS members US$71.20, List US$89, Order code AMSTEXT/29 Roberto Frigerio, University of Pisa, Italy

General Interest The theory of bounded cohomology, introduced by Gromov in the late 1980s, has had powerful applications in A Primer of geometric group theory and the geometry and topology of manifolds, and has been Mathematical Writing the topic of active research continuing Being a Disquisition to this day. This monograph provides a unified, self-contained on Having Your Ideas introduction to the theory and its applications, making it accessible to a student who has completed a first course in Recorded, Typeset, algebraic topology and manifold theory. The book can be used as a Published, Read, and source for research projects for master’s students, as a thorough Appreciated introduction to the field for graduate students, and as a valuable landmark text for researchers, providing both the details of the Second Edition theory of bounded cohomology and links of the theory to other closely related areas. Steven G. Krantz, Washington University, St. Louis, MO The first part of the book is devoted to settling the fundamental definitions of the theory, and to proving some of the (by now This is the second edition of a book originally published in 1997. classical) results on low-dimensional bounded cohomology and Today the internet virtually consumes all of our lives (especially on bounded cohomology of topological spaces. The second part the lives of writers). As both readers and writers, we are all describes applications of the theory to the study of the simplicial aware of blogs, chat rooms, and preprint servers. There are now volume of manifolds, to the classification of circle actions, to the electronic-only journals and print-on-demand books, Open Access analysis of maximal representations of surface groups, and to the journals and joint research projects such as MathOverflow—not study of flat vector bundles with a particular emphasis on the to mention a host of other new realities. It truly is a brave new possible use of bounded cohomology in relation with the Chern world, one that can be overwhelming and confusing. The truly conjecture. Each chapter ends with a discussion of further reading new feature of this second edition is an extensive discussion of that puts the presented results in a broader context. technological developments. Similar to the first edition, Krantz’s The author manages a near perfect equilibrium between necessary frank and straightforward approach makes this book particularly technicalities (always well motivated) and geometric intuition, suitable as a textbook for an undergraduate course. leading the readers from the first simple definition to the most Reviews and Endorsements of the First Edition: striking applications of the theory in 13 very pleasant chapters. This book can serve as an ideal textbook for a graduate topics Krantz, a prolific and distinguished mathematical author, course on the subject and become the much-needed standard discourses engagingly (yet seriously) on the art and etiquette reference on Gromov’s beautiful theory. of virtually all types of writing an academic mathematician is likely to encounter … Grammatical points, stylistic and typesetting —Michelle Bucher issues, and the correct and effective use of mathematical notation Contents: (Bounded) cohomology of groups; (Bounded) are handled deftly and with good humor … [Hopefully] senior cohomology of groups in low degree; Amenability; (Bounded) faculty will consider it mandatory reading for graduate students group cohomology via resolutions; Bounded cohomology of and even upper-division undergraduates. An enjoyable way to topological spaces; ℓ1-homology and duality; Simplicial volume; learn some fundamentals of good mathematical writing. Highly The proportionality principle; Additivity of the simplicial recommended. volume; Group actions on the circle; The Euler class of sphere —CHOICE bundles; Milnor-Wood inequalities and maximal representations; The bounded Euler class in higher dimensions and the Chern Well written in a lively style and will be found useful by anybody conjecture; Index; List of symbols; Bibliography. who is aware of the power and significance of writing in the mathematical profession. Mathematical Surveys and Monographs, Volume 227 —European Mathematical Society Newsletter December 2017, 191 pages, Hardcover, ISBN: 978-1-4704-4146-3, LC 2017023394, 2010 Mathematics Subject Classification: 18G60, Contents: The basics; Topics specific to the writing of mathematics; Exposition; Other types of writing; Books; Writing 20J06, 55N10, 57N65; 37E10, 37C85, 53C23, 57M07, 57R20, AMS with a computer; The world of high-tech publishing; Closing members US$92.80, List US$116, Order code SURV/227 thoughts; Bibliography; Index. December 2017, 243 pages, Softcover, ISBN: 978-1-4704-3658- 2, 2010 Mathematics Subject Classification: 00-01, 00-02, AMS members US$36, List US$45, Order code MBK/112

December 2017 Notices of the AMS 1343 New Publications Offered by the AMS

Nonsmooth Graduate Studies in Mathematics, Volume 187 Differential Geometry– January 2018, 368 pages, Hardcover, ISBN: 978-1-4704-2950-8, LC 2017028964, 2010 Mathematics Subject Classification: 58E12, An Approach Tailored 58E05; 46T10, 53C20, 55P62, AMS members US$66.40, List US$83, for Spaces with Ricci Order code GSM/187 Curvature Bounded from Below Number Theory Nicola Gigli, SISSA, Trieste, Italy

Contents: Introduction; The machinery Harmonic Maass of 퐿푝(픪)-normed modules; First order differential structure of Forms and Mock general metric measure spaces; Second order differential structure of RCD(퐾, ∞) spaces; Bibliography. Modular Forms: Memoirs of the American Mathematical Society, Volume 251, Theory and Number 1196 Applications December 2017, 161 pages, Softcover, ISBN: 978-1-4704-2765- 8, Individual member US$47, List US$78, Institutional member Kathrin Bringmann, University US$62, Order code MEMO/251/1196 of Cologne, Germany, Amanda Folsom, Amherst College, MA, Introduction to Global Ken Ono, Emory University, Atlanta, GA, and Larry Rolen, Analysis Trinity College, Dublin, Ireland

Minimal Surfaces in Modular forms and Jacobi forms play a central role in many areas Riemannian Manifolds of mathematics. Over the last 10–15 years, this theory has been extended to certain non-holomorphic functions, the so-called John Douglas Moore, University “harmonic Maass forms”. The first glimpses of this theory of California. Santa Barbara, CA appeared in Ramanujan’s enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions During the last century, global analysis he called “mock theta functions” which over eighty years later was one of the main sources of interaction were recognized as pieces of harmonic Maass forms. This book between geometry and topology. One contains the essential features of the theory of harmonic Maass might argue that the core of this subject is Morse theory, according forms and mock modular forms, together with a wide variety of to which the critical points of a generic smooth proper function on applications to algebraic number theory, , elliptic a manifold 푀 determine the homology of the manifold. curves, mathematical physics, quantum modular forms, and Morse envisioned applying this idea to the calculus of variations, representation theory. including the theory of periodic motion in classical mechanics, by Contents: Background: Elliptic functions; Theta functions and approximating the space of loops on 푀 by a finite-dimensional holomorphic Jacobi forms; Classical Maass forms; Harmonic manifold of high dimension. Palais and Smale reformulated Maass forms and mock modular forms: The basics; Differential Morse’s calculus of variations in terms of infinite-dimensional operators and mock modular forms; Examples of harmonic Maass manifolds, and these infinite-dimensional manifolds were found forms; Hecke theory; Zwegers’ thesis; Ramanujan’s mock theta useful for studying a wide variety of nonlinear PDEs. functions; Holomorphic projection; Meromorhic Jacobi forms; This book applies infinite-dimensional manifold theory to the Mock modular Eichler-shimura theory; Related automorphic Morse theory of closed geodesics in a Riemannian manifold. It forms; Applications: Partitions and unimodal sequences; then describes the problems encountered when extending this Asymptotics for coefficients of modular-type functions; Harmonic theory to maps from surfaces instead of curves. It treats critical Maass forms as arithmetic and geometric generating functions; point theory for closed parametrized minimal surfaces in a Shifted convolution 퐿-functions; Generalized Borcherds products; compact Riemannian manifold, establishing Morse inequalities for Elliptic curves over ℚ; Representation theory and mock modular perturbed versions of the energy function on the mapping space. It forms; Quantum modular forms; Representations of mock theta studies the bubbling which occurs when the perturbation is turned functions; Bibliography; Index. off, together with applications to the existence of closed minimal Colloquium Publications, Volume 64 surfaces. The Morse-Sard theorem is used to develop transversality theory for both closed geodesics and closed minimal surfaces. January 2018, approximately 390 pages, Hardcover, ISBN: 978- 1-4704-1944-8, 2010 Mathematics Subject Classification: 11F03, This book is based on lecture notes for graduate courses on 11F11, 11F27, 11F30, 11F37, 11F50, AMS members US$83.20, List “Topics in Differential Geometry”, taught by the author over US$104, Order code COLL/64 several years. The reader is assumed to have taken basic graduate courses in differential geometry and algebraic topology. Contents: Infinite-dimensional manifolds; Morse theory of geodesics; Topology of mapping spaces; Harmonic and minimal surfaces; Generic metrics; Bibliography; Index.

1344 Notices of the AMS Volume 64, Number 11 New AMS-Distributed Publications

Probability and Statistics at the University of Tokyo. This volume, the proceedings of the meeting, collects survey and research articles by international specialists in this fast-growing field. Spatially Independent This volume is recommended for researchers and graduate students interested in hyperbolic geometry, geometric group Martingales, theory, and low-dimensional topology. Intersections, and This item will also be of interest to those working in geometry and Applications topology. Published for the Mathematical Society of Japan by Kinokuniya, Pablo Shmerkin, Torcuato Di Tokyo, and distributed worldwide, except in Japan, by the AMS. Tella University, Buenos Aires, Advanced Studies in Pure Mathematics, Volume 73 Argentina, and CONICET, Buenos September 2017, 280 pages, Hardcover, ISBN: 978-4-86497-042-6, Aires, Argentina, and Ville 2010 Mathematics Subject Classification: 20F65, 57M50; 20E05, Suomala, University of Oulu, 20E08, 20F05, 20F28, 20F55, 37A20, 37F30, 37F50, 51M10, 57N16, Finland 57S05, 57S30, 58D05, 58D27, AMS members US$50.40, List US$63, Order code ASPM/73 Contents: Introduction; Notation; The setting; Hölder continuity of intersections; Classes of spatially independent martingales; A geometric criterion for Hölder continuity; Affine intersections Semisimple Lie and projections; Fractal boundaries and intersections with algebraic curves; Intersections with self-similar sets and measures; Algebras and Their Dimension of projections: applications of Theorem 4.4; Upper Classification Over bounds on dimensions of intersections; Lower bounds for the dimension of intersections, and dimension conservation; 푝-Adic Fields Products and convolutions of spatially independent martingales; Applications to Fourier decay and restriction; Bibliography. Torsten Schoeneberg, Camosun Memoirs of the American Mathematical Society, Volume 251, College, Victoria, BC, Canada Number 1195 This book gives a detailed structure December 2017, 100 pages, Softcover, ISBN: 978-1-4704-2688-0, theory for semisimple Lie algebras 2010 Mathematics Subject Classification: 28A75, 60D05; 28A78, over arbitrary fields of characteristic 0. 28A80, 42A38, 42A61, 60G46, 60G57, Individual member Starting from the well-known classification over algebraically US$47, List US$78, Institutional member US$62, Order code closed fields via root systems, the author mimics the language MEMO/251/1195 of reductive groups, so that part of his work can be seen as an introduction to a simpler version of Borel-Tits theory. But the author also expresses his results in the language of classical (matrix) algebra as well as Galois cohomology. In the second part of this book, the author focuses on 푝-adic fields New AMS-Distributed and achieves a complete classification of semisimple Lie algebras over them. This classification consists essentially of a list of so-called Satake-Tits diagrams, which extend the Dynkin diagrams Publications from the split case. Several instructive examples and historical notes supplement the text. This book can be used as the basis for a lecture on semisimple Lie algebras beyond the beginner’s level and Algebra and Algebraic as a reference for researchers. Geometry A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members Hyperbolic Geometry of the SMF receive a 30% discount from list. and Geometric Group Mémoires de la Société Mathématique de France, Number 151 Theory October 2017, 147 pages, Softcover, ISBN: 978-2-85629-859- 6, 2010 Mathematics Subject Classification: 17B05, 17B20, AMS Koji Fujiwara, Kyoto University, members US$41.60, List US$52, Order code SMFMEM/151 Japan, Sadayoshi Kojima, Tokyo Institute of Technology, Japan, and Ken’ichi Ohshika, Osaka University, Japan, Editors

The 7th Seasonal Institute of the Mathematical Society of Japan on Hyperbolic Geometry and Geometric Group Theory was held from July 30–August 5, 2014,

December 2017 Notices of the AMS 1345 New AMS-Distributed Publications

Geometry and Topology unified many different subjects of mathematics, including number theory, geometry, complex analysis, and discrete subgroups. The first book on icosahedron and the solution of equations of the The Bochner fifth degree showed closed relations between three seemingly different subjects: the symmetries of the icosahedron, the solution Technique in to fifth degree algebraic equations, and the differential equation of Differential Geometry hypergeometric functions. It was translated into English in 1888, four years after its original German version was published in 1884. Hung-Hsi Wu, University of It was followed by two volumes on elliptic modular functions by California, Berkeley Klein and Fricke and two more volumes on automorphic functions also by Klein and Fricke. This monograph is a detailed survey of an These four classic books (each volume available separately) are area of differential geometry surrounding vast generalizations of the first volume and contain the highly the Bochner technique. This is a technique original works of Poincaré and Klein on automorphic forms. They that falls under the general heading of have been very influential in the development of mathematics and “curvature and topology” and refers to a method initiated by are now available in English for the first time. These books contain in the 1940s for proving that on compact many original ideas, striking examples, explicit computations, and Riemannian manifolds, certain objects of geometric interest details which are not available anywhere else. They will be very (e.g., harmonic forms, harmonic spinor fields, etc.) must satisfy valuable references for people at all levels and allow the reader to additional differential equations when appropriate curvature see the unity of mathematics through the eyes of one of the most conditions are imposed. influential mathematicians with vision, Felix Klein. In 1953 K. Kodaira applied this method to prove the vanishing A publication of Higher Education Press (). Distributed in theorem for harmonic forms with values in a holomorphic vector North America by the American Mathematical Society. bundle. This theorem, which bears his name, was the crucial Classical Topics in Mathematics, Volume 1 step that allowed him to prove his famous imbedding theorem. Subsequently, the Bochner technique has been extended, on October 2017, 639 pages, Hardcover, ISBN: 978-7-04-047872- the one hand, to spinor fields and harmonic maps and, on the 3, 2010 Mathematics Subject Classification: 11-XX, 11F03, AMS other, to harmonic functions and harmonic maps on noncompact members US$71.20, List US$89, Order code CTM/1 manifolds. The last has led to the proof of rigidity properties of certain Kähler manifolds and locally symmetric spaces. This monograph gives a self-contained and coherent account of some of these developments, assuming the basic facts about Lectures on the Riemannian and Kähler geometry as well as the statement of Theory of Elliptic the Hodge theorem. The brief introductions to the elementary portions of spinor geometry and harmonic maps may be especially Modular Functions: useful to beginners. Second Volume This item will also be of interest to those working in analysis. A publication of Higher Education Press (Beijing). Distributed in Felix Klein and Robert Fricke North America by the American Mathematical Society. Translated by Arthur M. DuPre. Classical Topics in Mathematics, Volume 6 Felix Klein’s famous Erlangen program October 2017, 214 pages, Hardcover, ISBN: 978-7-04-047838- made the theory of group actions into a 9, 2010 Mathematics Subject Classification: 53-XX, 58-XX, AMS central part of mathematics. In the spirit members US$47.20, List US$59, Order code CTM/6 of this program, Klein set out to write a grand series of books which unified many different subjects of mathematics, including number theory, geometry, complex analysis, and discrete subgroups. The first book on icosahedron and the solution of equations of the Number Theory fifth degree showed closed relations between three seemingly different subjects: the symmetries of the icosahedron, the solution to fifth degree algebraic equations, and the differential equation of Lectures on the hypergeometric functions. It was translated into English in 1888, Theory of Elliptic four years after its original German version was published in 1884. It was followed by two volumes on elliptic modular functions by Modular Functions: Klein and Fricke and two more volumes on automorphic functions First Volume also by Klein and Fricke. These four classic books (each volume available separately) are Felix Klein and Robert Fricke vast generalizations of the first volume and contain the highly Translated by Arthur M. DuPre. original works of Poincaré and Klein on automorphic forms. They have been very influential in the development of mathematics and Felix Klein’s famous Erlangen program are now available in English for the first time. These books contain made the theory of group actions into a many original ideas, striking examples, explicit computations, and central part of mathematics. In the spirit details which are not available anywhere else. They will be very of this program, Klein set out to write a grand series of books which valuable references for people at all levels and allow the reader to

1346 Notices of the AMS Volume 64, Number 11 New AMS-Distributed Publications see the unity of mathematics through the eyes of one of the most Lectures on influential mathematicians with vision, Felix Klein. the Theory of A publication of Higher Education Press (Beijing). Distributed in North America by the American Mathematical Society. Automorphic Classical Topics in Mathematics, Volume 2 Functions: Second October 2017, 589 pages, Hardcover, ISBN: 978-7-04-047837- Volume 2, 2010 Mathematics Subject Classification: 11-XX, 11F03, AMS members US$71.20, List US$89, Order code CTM/2 Felix Klein and Robert Fricke Translated by Arthur M. DuPre.

Lectures on Felix Klein’s famous Erlangen program made the theory of group actions into a central part of the Theory of mathematics. In the spirit of this program, Klein set out to write Automorphic a grand series of books which unified many different subjects of mathematics, including number theory, geometry, complex Functions: First analysis, and discrete subgroups. Volume The first book on icosahedron and the solution of equations of the fifth degree showed closed relations between three seemingly Felix Klein and Robert Fricke different subjects: the symmetries of the icosahedron, the solution Translated by Arthur M. DuPre. to fifth degree algebraic equations, and the differential equation of hypergeometric functions. It was translated into English in 1888, Felix Klein’s famous Erlangen program four years after its original German version was published in 1884. made the theory of group actions into a central part of It was followed by two volumes on elliptic modular functions by mathematics. In the spirit of this program, Klein set out to write Klein and Fricke and two more volumes on automorphic functions a grand series of books which unified many different subjects also by Klein and Fricke. of mathematics, including number theory, geometry, complex These four classic books (each volume available separately) are analysis, and discrete subgroups. vast generalizations of the first volume and contain the highly The first book on icosahedron and the solution of equations of the original works of Poincaré and Klein on automorphic forms. They fifth degree showed closed relations between three seemingly have been very influential in the development of mathematics and different subjects: the symmetries of the icosahedron, the solution are now available in English for the first time. These books contain to fifth degree algebraic equations, and the differential equation of many original ideas, striking examples, explicit computations, and hypergeometric functions. It was translated into English in 1888, details which are not available anywhere else. They will be very four years after its original German version was published in 1884. valuable references for people at all levels and allow the reader to It was followed by two volumes on elliptic modular functions by see the unity of mathematics through the eyes of one of the most Klein and Fricke and two more volumes on automorphic functions influential mathematicians with vision, Felix Klein. also by Klein and Fricke. A publication of Higher Education Press (Beijing). Distributed in These four classic books (each volume available separately) are North America by the American Mathematical Society. vast generalizations of the first volume and contain the highly Classical Topics in Mathematics, Volume 4 original works of Poincaré and Klein on automorphic forms. They have been very influential in the development of mathematics and October 2017, 563 pages, Hardcover, ISBN: 978-7-04-047839- are now available in English for the first time. These books contain 6, 2010 Mathematics Subject Classification: 11-XX, 11F03, AMS many original ideas, striking examples, explicit computations, and members US$71.20, List US$89, Order code CTM/4 details which are not available anywhere else. They will be very valuable references for people at all levels and allow the reader to see the unity of mathematics through the eyes of one of the most influential mathematicians with vision, Felix Klein. A publication of Higher Education Press (Beijing). Distributed in North America by the American Mathematical Society. Classical Topics in Mathematics, Volume 3 October 2017, 539 pages, Hardcover, ISBN: 978-7-04-047840- 2, 2010 Mathematics Subject Classification: 11-XX, 11F03, AMS members US$71.20, List US$89, Order code CTM/3

December 2017 Notices of the AMS 1347 MATHEMATICS CALENDAR

As of the January 2018 issue of Notices, Mathematics Calendar will January 2018 be an online-only product. 3 – 5 UCLA Topology Workshop 2018 The AMS invites you to submit announcements of worldwide Location: University of California, Los Angeles, California. meetings and conferences of interest to the mathematical public, including ad hoc, local, or regional meetings, and URL: https://math.ucla.edu/~sucharit meetings and symposia devoted to specialized topics, as /TopologyWorkshop18 well as announcements of regularly scheduled meetings of 11 – 12 Two Nonlinear Days in Perugia on the Occasion of Patrizia national or international mathematical organizations to the interactive calendar at www.ams.org/cgi-bin/mathcal/ Pucci’s Sixty-fifth Birthday mathcal-submit.pl. Location: Dipartimento di Matematica e Informatica, Univer- Mathematics Calendar is available at: www.ams.org/ sity of Perugia, Italy. meetings/calendar/mathcal URL: www.dmi.unipg.it/nonlinearperugia2018 Any questions about or difficulties with submissions may be February 2018 directed to [email protected]. 12 – 16 Recent Advances in Hamiltonian Dynamics and Symplectic Topology November 2017 Location: Department of Mathematics ’Tullio Levi-Civita’, Uni- 16 – 17 Distributional Symmetries and Independences versity of Padua, Italy. Location: Institute of Mathematics of Bordeaux, Bordeaux, URL: events.math.unipd.it/hamschool2018 France. 20 – 24 Workshop in Nonlinear PDE’s and Geometric Analysis URL: sites.google.com/view/symmetries-meeting Location: Universidade Federal da Paraíba, João Pessoa, . /home URL: www.mat.ufpb.br/wenlu 17 – 19 2017 Midwest Geometry Conference Location: Kansas State University, Manhattan, Kansas. March 2018 URL: https://www.math.ksu.edu/events/conference 8 – 10 Latinx in the Mathematical Sciences Conference 2018 /2017_Midwest_Geometry/2017_Midwest_Geometry.html Location: Institute for Pure and Applied Mathematics (IPAM), UCLA, Los Angeles, California. December 2017 URL: www.ipam.ucla.edu/lat2018

4 – 6 GAP 2017: Geometry and Physics: Curvature Flows in Complex 19 – 24 Ischia Group Theory 2018 Geometry Location: Ischia (Naples), Italy. Location: Fields Institute, Toronto, Canada. URL: www.dipmat2.unisa.it/ischiagrouptheory URL: www.fields.utoronto.ca/activities/17-18 /gap2017 April 2018 8 – 10 Tech Topology Conference 2 – June 18 Doc-Course IEMath-Gr, IMUS, University of Cádiz and Location: Georgia Institute of Technology, Atlanta, Georgia. University of Málaga (Spain). April-June 2018 URL: ttc.gatech.edu Location: IEMath-Gr (Granada), IMUS (Sevilla), Universities of Cádiz and Málaga (Cádiz, Málaga), Spain. ∗9 – 11 International Conference on Algebra, Discrete Mathematics, URL: www.imus.us.es/DOC-COURSE18/en and Applications (ICADMA 2017) Location: Department of Mathematics, Dr. Babasaheb Ambed- 9 – 11 Algebraic Analysis in Honor of Pierre Schapira’s Seventy-fifth kar Marathwada University, Aurangabad 431004 India. Birthday URL: bamu.ac.in/int-conference-on-maths-2017 Location: Institut Henri Poincaré, Paris, France. /Home.aspx URL: monge.u-bourgogne.fr/gdito/pierre75

11 – 13 Risk Modeling, Management, and Mitigation in Health Sci- 26 – 27 International Conference on Frontiers in Industrial and ences Applied Mathematics Location: National Institute of Technology, Hamirpur, HP, In- Location: Centre de Recherches Mathématiques, Université de dia. Montréal Pavillon André-Aisenstadt 2920, Chemin de la tour, URL: www.fiam2018.org 5th floor Montréal (Québec) H3T 1J4 Canada. URL: www.crm.umontreal.ca/2017/Sante17 May 2018 12 – 14 Mini-Workshop on Log-Correlated Random Fields 25 – 28 Geometry and Topology of 3-Manifolds Workshop Location: Columbia University, New York, New York. Location: Okinawa Institute of Science and Technology (OIST), URL: www.math.columbia.edu/department/probability Okinawa, Japan. /log17/log17.html URL: groups.oist.jp/manifolds/workshop

1348 Notices of the AMS Volume 64, Number 11 Classified Advertising Positions available, items for sale, services available, and more

CALIFORNIA open. UCLA is an equal opportunity/ graduate and/or graduate courses, men- affirmative action employer. tor junior faculty, and participate in the governance of the Department, College, Institute for Pure and 00045 and University by serving on faculty com- Applied Mathematics mittees. University of California—Los Angeles Applications should be submitted on- INDIANA line through www.mathjobs.org/jobs/ The Institute for Pure and Applied Math- jobs/10781 and should include (1) the ematics (IPAM) at UCLA is seeking an As- Purdue University AMS cover sheet for academic employ- sociate Director (AD), to begin a two-year Department: Mathematics ment, (2) a curriculum vitae, (3) a research appointment on August 1, 2018. Position Title: Andris A. Zoltners statement, and (4) four letters of recom- The AD is expected to be an active and Professor mendation. Preferably one of the letters established research mathematician or will discuss the candidate’s teaching. scientist in a related field, with experience The Department of Mathematics invites Reference letter writers should be asked in conference organization. The primary applications for an appointment at the to submit their letters online through responsibility of the AD will be running rank of tenured full professor to fill the www.mathjobs.org. individual programs in coordination with endowed Andris A. Zoltners Professorship Direct all inquiries to mathhead@pur- the organizing committees. The selected in Mathematics. A PhD (or its equivalent) due.edu. Screening of applications will in mathematics or a closely related field candidate will be encouraged to continue begin October 30, 2017, and continue until is required. We are expecting applications filled. Some offers may be made before the his or her personal research program from candidates with an outstanding re- end of January 2018. within the context of the responsibilities cord of research accomplishments, inter- For information about our department, to the institute. nationally recognized stature, credentials see www.math.purdue.edu/. A back- For a detailed job description and ap- suitable for immediate nomination as a ground check will be required for employ- plication instructions, go to www.ipam. Distinguished Professor, and great po- ment in these positions. ucla.edu/5nhZK. Applications will re- tential for future work. We will consider Purdue University’s Department of ceive fullest consideration if received by applications in any area of mathematics. Mathematics is committed to advancing February 15, 2018, but we will accept ap- Duties: Conduct research in mathemat- diversity in all areas of faculty effort, plications as long as the position remains ics, interact with faculty, teach under- including scholarship, instruction, and

Suggested uses for classified advertising are positions available, books or lecture notes for sale, books being sought, exchange or rental of houses, and typing services. The publisher reserves the right to reject any advertising not in keeping with the publication's standards. Acceptance shall not be construed as approval of the accuracy or the legality of any advertising. The 2017 rate is $3.50 per word with a minimum two-line headline. No discounts for multiple ads or the same ad in consecutive issues. For an additional $10 charge, announcements can be placed anonymously. Correspondence will be forwarded. Advertisements in the “Positions Available” classified section will be set with a minimum one-line headline, consisting of the institution name above body copy, unless additional headline copy is specified by the advertiser. Headlines will be centered in boldface at no extra charge. Ads will appear in the language in which they are submitted. There are no member discounts for classified ads. Dictation over the telephone will not be accepted for classified ads. Upcoming deadlines for classified advertising are as follows: September 28, 2017; January 2018—October 31, 2017; February 2018—November 23, 2017; March 2018—January 2, 2018; April 2018—January 30, 2018; May 2018—March 2, 2018; June/July 2018—April 27, 2018. US laws prohibit discrimination in employment on the basis of color, age, sex, race, religion, or national origin. “Positions Available” advertisements from institutions outside the US cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to US laws. Details and specific wording may be found on page 1373 (vol. 44). Situations wanted advertisements from involuntarily unemployed mathematicians are accepted under certain conditions for free publication. Call toll-free 800-321-4AMS (321-4267) in the US and Canada or 401-455-4084 worldwide for further information. Submission: Promotions Department, AMS, P.O. Box 6248, Providence, Rhode Island 02904; or via fax: 401-331-3842; or send email to [email protected]. AMS location for express delivery packages is 201 Charles Street, Providence, Rhode Island 02904. Advertisers will be billed upon publication.

December 2017 Notices of the AMS 1349 Classified Advertising engagement. Candidates should address engagement. Candidates should address KANSAS at least one of these areas in their cover at least one of these areas in their cover letter, indicating their past experiences, letter, indicating their past experiences, University of Kansas current interests or activities, and/or fu- current interests or activities, and/or fu- Department of Mathematics ture goals to promote a climate that values ture goals to promote a climate that values diversity and inclusion. diversity and inclusion. The Department of Mathematics at the Purdue University is an EOE/AA em- University of Kansas invites applications ployer. All individuals, including minori- Purdue University is an EOE/AA em- for a tenure-track, Assistant Professor ties, women, individuals with disabilities, ployer. All individuals, including minori- position in Geometry expected to begin as and veterans are encouraged to apply. ties, women, individuals with disabilities, early as August 18, 2018. Candidates must 00043 and veterans are encouraged to apply. demonstrate an outstanding record of re- search and must be strongly committed to 00042 Purdue University excellence in teaching. Requirements for the position include a PhD in mathemat- Department of Mathematics Purdue University Tenure track: Assistant Professor ics and outstanding publication record in Department: Mathematics of Mathematics Differential Geometry and/or Algebraic Position Title: Golomb Visiting Geometry, with an emphasis on Geometry. The Mathematics Department at Purdue Assistant Professor and Zoltners For a complete announcement and to University invites applications for up Visiting Assistant Professor apply online, go to employment.ku.edu/ to two possible appointments in math- academic/10039BR . At least four recom- ematics to begin August 2018. These These three-year positions intended for mendation letters (teaching ability must be addressed in at least one letter) should appointments will be at the level of as- new and very recent PhDs will commence sistant professor. Appointments will be be submitted electronically to www.math- August 2018 and are open to mathemati- made based on demonstrated research jobs.org/jobs/jobs/ , position #10761. and teaching qualifications. PhD (or its cians who demonstrate exceptional re- Initial review of applications will begin equivalent) in mathematics or a closely search promise and a strong teaching November 1, 2017, and continue as long as related field is required. Preference will record. Ph.D. (or its equivalent) in math- needed to identify a qualified pool. be given to outstanding applicants in the ematics or closely related field by August KU is an EO/AAE. All qualified ap- areas of Analysis and Geometry (including 14, 2018 is required. Applicants should plicants will receive consideration for stochastic analysis/probability, harmonic have research interests in common with employment without regard to race, color, analysis, partial differential equations, Purdue faculty. Duties include teaching religion, sex (including pregnancy), age, complex analysis, and symplectic/dif- national origin, disability, genetic infor- undergraduate and graduate mathemat- ferential geometry), Algebra (including mation or protected Veteran status. commutative algebra, algebraic topology, ics courses. 00044 algebraic geometry, automorphic forms Applications should be submitted on- and number theory), and Computational line through www.mathjobs.org and and Applied Mathematics (including ap- should include (1) the AMS cover sheet MASSACHUSETTS plied, numerical, and computational anal- for academic employment, (2) a curricu- ysis, the modeling of physical/biological lum vita, (3) a research statement, and (4) systems, and inverse problems). Northeastern University three letters of recommendation, one of Duties: Conduct research in mathemat- College of Science, Department of ics. Teach undergraduate and/or graduate which discusses the candidate’s teaching Mathematics and the College of mathematics courses to a diverse student qualifications. Reference letter writers Computer and Information Science body and supervise graduate students. should be asked to submit their letters on- Joint Tenured/Tenure-Track Position, Senior faculty will also mentor junior line through www.mathjobs.org/jobs/ All Levels faculty and participate in the governance jobs/10779 . Direct all inquiries to math- of the department, the College of Science, [email protected]. Screening of applica- The College of Computer and Information Science and the Department of Mathemat- and the University by serving on faculty tions will begin November 1, 2017 and committees. ics, in the College of Science, at Northeast- continue until filled. Some offers will be Applications should be submitted on- ern University invite applications for an made before the end of January 2018. For line through www.mathjobs.org/jobs/ open tenure-track/tenured faculty posi- jobs/10778 and should include (1) the information about our department, see tion at all levels, in Mathematics and Data AMS cover sheet for academic employ- www.math.purdue.edu/. A background Science, beginning in Fall 2018. ment, (2) a curriculum vitae, (3) a research check will be required for employment in Appointments will be based on ex- statement, and (4) four letters of recom- these positions. ceptional research contributions at the mendation, one of which discusses the Purdue University’s Department of interface between Mathematics and Com- candidate’s teaching qualifications. Ref- puter Science, combined with a strong Mathematics is committed to advancing erence letter writers should be asked to commitment and demonstrated success submit their letters online through www. diversity in all areas of faculty effort, in teaching. mathjobs.org. Direct all inquiries to including scholarship, instruction, and Candidates will be considered from [email protected]. engagement. Candidates should address all areas in Computer and Data Science, Applications are considered on a con- at least one of these areas in their cover Machine Learning, Discrete and Compu- tinuing basis but candidates are urged letter, indicating their past experiences, tational Mathematics, Probability and to apply by November 1, 2017. For more current interests or activities, and/or fu- Statistics, and Topological Data Analysis. information about our department, see ture goals to promote a climate that values Qualifications: A PhD in Computer Sci- www.math.purdue.edu/. A background diversity and inclusion. ence, Mathematics or a closely related check will be required for employment in field to one of the above-listed areas of this position. Purdue University is an EOE/AA em- expertise by the start date is required. Suc- Purdue University’s Department of ployer. All individuals, including minori- cessful candidates are expected to have Mathematics is committed to advancing ties, women, individuals with disabilities, or to develop an independently funded diversity in all areas of faculty effort, and veterans are encouraged to apply. research program of international caliber including scholarship, instruction, and 00041 and demonstrated evidence of excellent

1350 Notices of the AMS Volume 64, Number 11 Classified Advertising teaching in undergraduate and graduate probability theory and stochastic analysis Associate, or Full Professor in all areas courses. Qualified candidates should be or mathematical finance as well as a ca- of mathematics. committed to fostering diverse and inclu- pacity for effective teaching. The required •• Full-time tenure-track or tenured ap- sive environments as well as to promoting application materials are as follows: (a) pointments at the level of Assistant, experiential learning, which are central to AMS Cover Sheet, (b) curriculum vitae Associate, or Full Professor in all areas a Northeastern University education. (including a list of publications), (c) a re- of statistics. These positions are part Additional Information: Northeastern search statement (not to exceed five pages of a University-wide cluster hiring University is home to 35,000 full- and part- in length), (d) four letters of recommenda- effort in statistics, with particular time degree students and to the nation’s tion, one of which evaluates teaching and emphasis in mathematics, computer premier cooperative education program. preferably describes quantitative evidence science, and bioengineering. Successful candidates will have strong interdisci- The past decade has witnessed a dramatic of teaching ability, and (e) a teaching state- plinary interests. increase in Northeastern’s international ment (not to exceed two pages in length). reputation for research and innovative •• Three-year Burgess, Tucker, and Wylie educational programs. A heightened focus The Teaching Assistant Professorship Assistant Professor Lecturer positions. (code MFTAP) has a teaching load of 3–3 on interdisciplinary research and scholar- Please see our website at www.math. ship is driving a faculty hiring initiative and assists the program director and staff utah.edu/positions for information at Northeastern, advancing its position for the Department's Mathematical Fi- regarding available positions and applica- amongst the nation’s top research uni- nance Master's Degree program, finmath. tion requirements. Applications must be versities. The College of Computer and rutgers.edu. Candidates for this position completed through www.mathjobs.org/ Information Science and the College of should have received a PhD degree and jobs/Utah. Review of complete applica- Science have been major participants in show outstanding evidence of teaching tions for tenure-track positions will begin this initiative and will continue their ef- ability in probability theory, stochastic on October 23, 2017, and will continue forts this year, with additional interdisci- analysis, and mathematical finance. The until the positions are filled. Completed plinary searches ongoing in related areas. required application materials are as fol- applications for postdoctoral positions re- For more information about the College of lows: (a) AMS Cover Sheet, (b) curriculum ceived before January 1, 2018, will receive Computer and Information Science, please vitae (including a list of publications full consideration. visit www.ccis.northeastern.edu, and and teaching experience), (c) a summary The University of Utah is an Equal Op- for the College of Science, please visit of research (not to exceed five pages in portunity/Affirmative Action employer www.northeastern.edu/cos/. length), (d) four letters of recommenda- and educator. Minorities, women, veterans, Additional information and instructions tion, at least one of which evaluates teach- and those with disabilities are strongly en- for submitting application materials may ing thoroughly and preferably describes couraged to apply. Veterans’ preference is be found at the following web site: neu. quantitative evidence of teaching ability, extended to qualified veterans. Reasonable disability accommodations will be provided peopleadmin.com/postings/search. and (e) a teaching statement (not to exceed with adequate notice. For additional infor- Screening of applications begins imme- two pages in length). diately. For full consideration, application mation about the University’s commitment All application materials for either po- materials should be received by December to equal opportunity and access see: www. sition should be submitted electronically 1, 2017. However, applications will be ac- utah.edu/nondiscrimination/. through the AMS website, mathjobs.org . cepted until the search is completed. 00029 Northeastern University is an Equal Review of applications begins December Opportunity, Affirmative Action Educa- 1, 2017, and continues until any potential tional Institution and Employer, Title IX openings are filled. TAIWAN University. All qualified applicants will Rutgers, the State University of New Jersey, is an Equal Opportunity / Affirma- receive consideration for employment National Chengchi University without regard to race, color, religion, sex, tive Action Employer. Qualified applicants Open Faculty Positions (Tenure Track) national origin, disability status, protected will be considered for employment with- veteran status, or any other characteristic out regard to race, creed, color, religion, The Department of Mathematical Sci- protected by the law. Northeastern Univer- sex, sexual orientation, gender identity ences at National Chengchi University in sity is an E-Verify Employer. or expression, national origin, disability Taipei, Taiwan anticipates openings for 00047 status, genetic information, protected vet- several tenure-track faculty positions. The eran status, military service or any other candidate must hold a doctoral degree category protected by law. As an institu- in (Applied) Mathematics and be able to NEW JERSEY tion, we value diversity of background communicate in Chinese and English. For and opinion, and prohibit discrimination more information, please visit www.math. Rutgers University—New Brunswick or harassment on the basis of any legally nccu.edu.tw/app/news.php?Sn=718. Mathematics Department protected class in the areas of hiring, 00015 recruitment, promotion, transfer, demo- (This is a supplement to the add posted tion, training, compensation, pay, fringe in the October Notices.) benefits. Subject to availability of funding, the 00046 Department expects to have two open positions in probability theory and sto- chastic analysis or mathematical finance with start date in Fall 2018: (1) three-year, UTAH non-tenure track, nonrenewable Postdoc- toral Assistant Professorship and (2) non- University of Utah tenure track, renewable contract Teaching Department of Mathematics Assistant Professorship. The Postdoctoral Assistant Professor- The Department of Mathematics at the ship (code MFP) has a teaching load of 2–1 University of Utah invites applications for and candidates for this position should the following positions: have received a PhD degree, show out- •• Full-time tenure-track or tenured ap- standing promise of research ability in pointments at the level of Assistant,

December 2017 Notices of the AMS 1351 MEETINGS & CONFERENCES OF THE AMS DECEMBER TABLE OF CONTENTS

The Meetings and Conferences section of The most up-to-date meeting and confer- necessary to submit an electronic form, the Notices gives information on all AMS ence information can be found online at: although those who use L ATEX may submit meetings and conferences approved by www.ams.org/meetings/. abstracts with such coding, and all math press time for this issue. Please refer to Important Information About AMS displays and similarily coded material the page numbers cited on this page for Meetings: Potential organizers, (such as accent marks in text) must more detailed information on each event. speakers, and hosts should refer to be typeset in LATEX. Visit www.ams.org/ Invited Speakers and Special Sessions are page 75 in the January 2017 issue of the cgi-bin/abstracts/abstract.pl/. Ques- listed as soon as they are approved by the Notices for general information regard- tions about abstracts may be sent to abs- cognizant program committee; the codes ing participation in AMS meetings and [email protected]. Close attention should be listed are needed for electronic abstract conferences. paid to specified deadlines in this issue. submission. For some meetings the list Abstracts: Speakers should submit ab- Unfortunately, late abstracts cannot be may be incomplete. Information in this stracts on the easy-to-use interactive accommodated. A issue may be dated. Web form. No knowledge of LTEX is

MEETINGS IN THIS ISSUE

–––––––– 2018 –––––––– –––––––– 2019 –––––––– January 16–19 Baltimore, Maryland p. 1363 January 10–13 San Diego, California p. 1353 March 15–17 Auburn, Alabama p. 1364 March 17–18 Columbus, Ohio p. 1356 March 22–24 Honolulu, Hawaii p. 1364 April 14–15 Nashville, Tennesse p. 1358 June 10–13 Quy Nhon City, Vietnam p.1364 October 12–13 Binghamton, New York p. 1364 April 14–15 Portland, Oregon p. 1359 November 2–3 Gainesville, Florida p. 1364 April 21–22 Boston, Massachusetts p. 1360 June 11–14 Shanghai, People's Republic –––––––– 2020 –––––––– of China p. 1361 January 15–18 Denver, Colorado p. 1365 September 29–30 Newark, Delaware p. 1362 –––––––– 2021 –––––––– October 6–7 Fayetteville, Arkansas p. 1362 January 6–9 Washington, DC p. 1365 October 20–21 Ann Arbor, Michigan p. 1362 October 27–28 San Francisco, California p. 1363

See www.ams.org/meetings/ for the most up-to-date information on the meetings and conferences that we offer.

ASSOCIATE SECRETARIES OF THE AMS Central Section: Georgia Benkart, University of Wisconsin- Southeastern Section: Brian D. Boe, Department of Mathemat- Madison, Department of Mathematics, 480 Lincoln Drive, ics, University of Georgia, 220 D W Brooks Drive, Athens, GA Madison, WI 53706-1388; email: [email protected]; 30602-7403, email: [email protected]; telephone: 706-542- telephone: 608-263-4283. 2547. Eastern Section: Steven H. Weintraub, Department of Math- Western Section: Michel L. Lapidus, Department of Mathemat- ematics, Lehigh University, Bethlehem, PA 18015-3174; email: ics, University of California, Surge Bldg., Riverside, CA 92521- [email protected]; telephone: 610-758-3717. 0135; email: [email protected]; telephone: 951-827-5910.

1352 NOTICES OF THE AMS VOLUME 64, NUMBER 11 MEETINGS & CONFERENCES

Meetings & Conferences of the AMS

IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program informa- tion with links to the abstract for each talk can be found on the AMS website. See www.ams.org/meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL .

Moon Duchin, Tufts University, Political Geometry: San Diego, California Voting districts,”compactness,” and ideas about fairness (MAA-AMS-SIAM Gerald and Judith Porter Public Lecture). San Diego Convention Center and André Neves, University of Chicago, Minimal surfaces, San Diego Marriott Hotel and Marina volume spectrum, and Morse index (AMS-MAA Invited Ad- dress). January 10–13, 2018 Jill Pipher, Brown University, Nonsmooth boundary Wednesday – Saturday value problems (AWM-AMS Noether Lecture). AMS Invited Addresses Meeting #1135 Federico Ardila, San Francisco State University, Alge- Joint Mathematics Meetings, including the 124th Annual braic structures on polytopes. Meeting of the AMS, 101st Annual Meeting of the Math- Robert L. Bryant, Duke University, The Concept of Holo- ematical Association of America (MAA), annual meetings nomy—Its History and Recent Developments (AMS Retiring of the Association for Women in Mathematics (AWM) and Presidential Address). the National Association of Mathematicians (NAM), and the Ruth Charney, Brandeis University, Searching for Hy- winter meeting of the Association of Symbolic Logic (ASL), perbolicity. with sessions contributed by the Society for Industrial and , Harvard University, Privacy in the Land Applied Mathematics (SIAM). of Plenty (AMS Josiah Willard Gibbs Lecture). Associate secretary: Georgia Benkart Dana Randall, Georgia Institute of Technology, Emer- Announcement issue of Notices: October 2017 gent phenomena in random structures and algorithms. Program first available on AMS website: To be announced Edriss S. Titi, Texas A&M University; and The Weizmann Program issue of electronic Notices: To be announced Institute of Science, The Navier-Stokes, Euler and Related Issue of Abstracts: Volume 39, Issue 1 Equations. Avi Wigderson, Institute for Advanced Study, Alternate Deadlines Minimization and Scaling algorithms: theory, applications and connections across mathematics and computer science For organizers: Expired (AMS Colloquium Lectures: Lecture I). For abstracts: Expired Avi Wigderson, Institute for Advanced Study, Proving algebraic identities (AMS Colloquium Lectures: Lecture II). The scientific information listed below may be dated. Avi Wigderson, Institute for Advanced Study, Proving For the latest information, see www.ams.org/amsmtgs/ analytic inequalities (AMS Colloquium Lectures: Lecture national.html. III).

Joint Invited Addresses AMS Special Sessions Gunnar Carlsson, Stanford University, Topological If you are volunteering to speak in a Special Ses- Modeling of Complex Data (AMS-MAA Invited Address). sion, you should send your abstract as early as pos-

December 2017 Notices of the AMS 1353 MEETINGS & CONFERENCES sible via the abstract submission form found at Beyond Planarity: Crossing Numbers of Graphs (a jointmathematicsmeetings.org/meetings/abstracts/ Mathematics Research Communities Session), Axel Brandt, abstract.pl?type=. Davidson College, Garner Cochran, University of South Carolina, and Sarah Loeb, College of William and Mary. Some sessions are cosponsored with other organiza- Bifurcations of Difference Equations and Discrete Dy- tions. These are noted within the parenthesis at the end namical Systems, Arzu Bilgin and Toufik Khyat, Univer- of each listing, where applicable. sity of Rhode Island. A Showcase of Number Theory at Liberal Arts Colleges, Boundaries for Groups and Spaces, Joseph Maher, Adriana Salerno, Bates College, and Lola Thompson, CUNY College of Staten Island, and Genevieve Walsh, Oberlin College. Tufts University. Accelerated Advances in Mathematical Fractional Pro- Combinatorial Commutative Algebra and Polytopes, gramming, Ram Verma, International Publications USA, Robert Davis, Michigan State University, and Liam Solus, and Alexander Zaslavski, Israel Institute of Technology. KTH Royal Institute of Technology. Advances in Applications of Differential Equations to Combinatorics and Geometry, Federico Ardila, San Disease Modeling, Libin Rong, Oakland University, Elissa Francisco State University, Anastasia Chavez, MSRI and Schwartz, Washington State University, and Naveen K. University of California, Davis, and Laura Escobar, Uni- Vaidya, San Diego State University. versity of Illinois Urbana Champaign. Advances in Difference, Differential, and Dynamic Equa- Commutative Algebra in All Characteristics, Neil Ep- tions with Applications, Elvan Akin, Missouri University stein, George Mason University, Karl Schwede, University S&T, and John Davis, Baylor University. of Utah, and Janet Vassilev, University of New Mexico. Advances in Operator Algebras, Marcel Bischoff, Computational Combinatorics and Number Theory, Vanderbilt University, Ian Charlesworth, University Jeremy F. Alm, Lamar University, and David Andrews of California, Los Angeles, Brent Nelson, University of and Rob Hochberg, University of Dallas. California, Berkeley, and Sarah Reznikoff, Kansas State Connections in Discrete Mathematics: Graphs, Hyper- University. graphs, and Designs, Amin Bahmanian, Illinois State Uni- Advances in Operator Theory, Operator Algebras, and versity, and Theodore Molla, University Illinois Urbana- Operator Semigroups, Asuman G. Aksoy, Claremont Champaign. McKenna College, Zair Ibragimov, California State Univer- Differential Geometry, Vincent B. Bonini and Joseph E. sity, Fullerton, Marat Markin, California State University, Borzellino, Cal Poly San Luis Obispo, Bogdan D. Suceava, Fresno, and Ilya Spitkovsky, New York University, Abu California State University, Fullerton, and Guofang Wei, Dhabi. University of California, Santa Barbara. Algebraic, Analytic, and Geometric Aspects of Inte- Diophantine Approximation and Analytic Number The- grable Systems, Painlevé Equations, and Random Matrices, ory in Honor of Jeffrey Vaaler, Shabnam Akhtari, Univer- Vladimir Dragovic, University of Texas at Dallas, Anton sity of Oregon, Lenny Fukshansky, Claremont McKenna Dzhamay, University of Northern Colorado, and Sevak College, and Clayton Petsche, Oregon State University. Mkrtchyan, University of Rochester. Discrete Dynamical Systems and Applications, E. Ca- Algebraic, Discrete, Topological and Stochastic Ap- bral Balreira, Saber Elaydi, and Eddy Kwessi, Trinity proaches to Modeling in Mathematical Biology, Olcay University. Akman, Illinois State University, Timothy D. Comar, Discrete Neural Networking and Applications, Murat Benedictine University, Daniel Hrozencik, Chicago State Adivar, Fayetteville State University, Michael A. Radin, University, and Raina Robeva, Sweet Briar College. Rochester Institute of Technology, and Youssef Raffoul, Alternative Proofs in Mathematical Practice, John W. University of Dayton. Dawson, Jr., Pennsylvania State University, York. Analysis of Fractional, Stochastic, and Hybrid Dynamic Dynamical Algebraic Combinatorics, James Propp, Systems, John R. Graef, University of Tennessee at Chat- University of Massachusetts, Lowell, Tom Roby, Univer- tanooga, Gangaram S. Ladde, University of South Florida, sity of Connecticut, Jessica Striker, North Dakota State and Aghalaya S. Vatsala, University of Louisiana at La- University, and Nathan Williams, University of California fayette. Santa Barbara. Analysis of Nonlinear Partial Differential Equations and Dynamical Systems with Applications to Mathematical Applications, Tarek M. Elgindi, University of California, Biology, Guihong Fan, Columbus State University, Jing Li, San Diego, and Edriss S. Titi, Texas A&M University and California State University Northridge, and Chunhua Shan, Weizmann Institute of Science. University of Toledo. Applied and Computational Combinatorics, Torin Dynamical Systems: Smooth, Symbolic, and Measurable Greenwood, Georgia Institute of Technology, and Jay (a Mathematics Research Communities Session), Kathryn Pantone, Dartmouth College. Lindsey, Boston College, Scott Schmieding, Northwestern Arithmetic Dynamics, Robert L. Benedetto, Amherst University, and Kurt Vinhage, University of Chicago. College, Benjamin Hutz, Saint Louis University, Jamie Emergent Phenomena in Discrete Models, Dana Randall, Juul, Amherst College, and Bianca Thompson, Harvey Georgia Institute of Technology, and Andrea Richa, Ari- Mudd College. zona State University.

1354 Notices of the AMS Volume 64, Number 11 MEETINGS & CONFERENCES

Emerging Topics in Graphs and Matrices, Sudipta Mal- Mathematical Modeling and Analysis of Infectious Dis- lik, Northern Arizona University, Keivan Hassani Mon- eases, Kazuo Yamazaki, University of Rochester. fared, University of Calgary, and Bryan Shader, University Mathematical Modeling of Natural Resources, Shandelle of Wyoming. M. Henson, Andrews University, and Natali Hritonenko, Ergodic Theory and Dynamical Systems–to Celebrate Prairie View A&M University. the Work of Jane Hawkins, Julia Barnes, Western Carolina Mathematical Modeling, Analysis and Applications in University, Rachel Bayless, Agnes Scott College, Emily Population Biology, Yu Jin, University of Nebraska-Lincoln, Burkhead, Duke University, and Lorelei Koss, Dickinson and Ying Zhou, Lafayette College. College. Mathematical Problems in Ocean Wave Modeling and Extremal Problems in Approximations and Geometric Fluid Mechanics, Christopher W. Curtis, San Diego State Function Theory, Ram Mohapatra, University of Central University, and Katie Oliveras, Seattle University. Florida. Mathematical Relativity and Geometric Analysis, James Financial Mathematics, Actuarial Sciences, and Re- Dilts and Michael Holst, University of California, San lated Fields, Albert Cohen, Michigan State University, Diego. Nguyet Nguyen, Youngstown State University, Oana Mathematics Research from the SMALL Undergradu- Mocioalca, Kent State University, and Thomas Wakefield, ate Research Program, Colin Adams, Frank Morgan, and Youngstown State University. Cesar E. Silva, Williams College. Fractional Difference Operators and Their Application, Mathematics of Gravitational Wave Science, Andrew Christopher S. Goodrich, Creighton Preparatory School, Gillette and Nikki Holtzer, University of Arizona. and Rajendra Dahal, Coastal Carolina University. Mathematics of Quantum Computing and Topological Free Convexity and Free Analysis, J. William Helton, Phases of Matter, Paul Bruillard, Pacific Northwest Na- University of California, San Diego, and Igor Klep, Uni- tional Laboratory, David Meyer, University of California versity of Auckland. San Diego, and Julia Plavnik, Texas A&M University. Geometric Analysis, Davi Maximo, University of Penn- Metric Geometry and Topology, Christine Escher, Or- egon State University, and Catherine Searle, Wichita State sylvania, Lu Wang, University of Wisconsin-Madison, and University. Xin Zhou, University of California Santa Barbara. Modeling in Differential Equations - High School, Two- Geometric Analysis and Geometric Flows, David Glick- Year College, Four-Year Institution, Corban Harwood, enstein, University of Arizona, and Brett Kotschwar, George Fox University, William Skerbitz, Wayzata High Arizona State University. School, Brian Winkel, SIMIODE, and Dina Yagodich, Fred- History of Mathematics, Sloan Despeaux, Western Caro- erick Community College. lina University, Jemma Lorenat, Pitzer College, Clemency Multi-scale Modeling with PDEs in Computational Science Montelle, University of Canterbury, Daniel Otero, Xavier and Engineering:Algorithms, Simulations, Analysis, and Ap- University, and Adrian Rice, Randolph-Macon College. plications, Salim M. Haidar, Grand Valley State University. Homotopy Type Theory (a Mathematics Research Com- Network Science, David Burstein, Swarthmore College, Simon Cho munities Session), , University of Michigan, Franklin Kenter, United States Naval Academy, and Feng Liron Cohen, Cornell University, and Edward Morehouse, Shi, University of North Carolina at Chapel Hill. Wesleyan University. New Trends in Celestial Mechanics, Richard Montgom- If You Build It They Will Come: Presentations by Scholars ery, University of California Santa Cruz, and Zhifu Xie, in the National Alliance for Doctoral Studies in the Math- University of Southern Mississippi. ematical Sciences, David Goldberg, Purdue University, and Nilpotent and Solvable Geometry,I, Michael Jablonski, Phil Kutzko, . University of Oklahoma, Megan Kerr, Wellesley College, Interactions of Inverse Problems, Signal Processing, and and Tracy Payne, Idaho State University. Imaging, M. Zuhair Nashed, University of Central Florida, Noncommutative Algebras and Noncommutative Invari- Willi Freeden, University of Kaiserslautern, and Otmar ant Theory, Ellen Kirkman, Wake Forest University, and Scherzer, University of . James Zhang, University of Washington. Markov Chains, Markov Processes and Applications, Nonlinear Evolution Equations of Quantum Physics and Alan Krinik and Randall J. Swift, California State Poly- Their Topological Solutions, Stephen Gustafson, University technic University. of British Columbia, Israel Michael Sigal, University of Mathematical Analysis and Nonlinear Partial Differen- Toronto, and Avy Soffer, Rutgers University. tial Equations, Hongjie Dong, Brown University, Peiyong Novel Methods of Enhancing Success in Mathematics Wang, Wayne State University, and Jiuyi Zhu, Louisiana Classes, Ellina Grigorieva, Texas Womans University, and State University. Natali Hritonenko, Prairie View A&M University. Mathematical Fluid Mechanics: Analysis and Applica- Open and Accessible Problems for Undergraduate tions, Zachary Bradshaw and Aseel Farhat, University Research, Michael Dorff, Brigham Young University, Al- of Virginia. lison Henrich, Seattle University, and Nicholas Scoville, Mathematical Information in the Digital Age of Science, Ursinus College. Patrick Ion, University of Michigan, Olaf Teschke, zbMath Operators on Function Spaces in One and Several Vari- Berlin, and Stephen Watt, University of Waterloo. ables, Catherine Bénéteau, University of South Florida,

December 2017 Notices of the AMS 1355 MEETINGS & CONFERENCES and Matthew Fleeman and Constanze Liaw, Baylor Uni- Structure and Representations of Hopf Algebras: a Ses- versity. sion in Honor of Susan Montgomery, Siu-Hung Ng, Louisi- Orthogonal Polynomials and Applications, Abey Lopez- ana State University, and Lance Small and Henry Tucker, Garcia, University of South Alabama, and Xiang-Sheng University of California, San Diego. Wang, University of Louisiana at Lafayette. Theory, Practice, and Applications of Graph Clustering, Orthogonal Polynomials, Quantum Probability, and David Gleich, Purdue University, and Jennifer Webster Stochastic Analysis, Julius N. Esunge, University of Mary and Stephen J. Young, Pacific Northwest National Labo- Washington, and Aurel I. Stan, Ohio State University. ratory. Quantum Link Invariants, , and Topological Data Analysis, Henry Adams, Colorado Low-dimensional Manifolds, Diana Hubbard, University State University, Gunnar Carlsson, Stanford University, of Michigan, and Christine Ruey Shan Lee, University of and Mikael Vejdemo-Johansson, CUNY College of Staten Texas at Austin. Island. Quaternions, Terrence Blackman, Medgar Evers Col- Topological Graph Theory: Structure and Symmetry, lege, City University of New York, and Johannes Familton Jonathan L. Gross, Columbia University, and Thomas W. and Chris McCarthy, Borough of Manhattan Community Tucker, Colgate University. College, City University of New York. Visualization in Mathematics: Perspectives of Mathema- Recent Trends in Analysis of Numerical Methods of ticians and Mathematics Educators, Karen Allen Keene, Partial Differential Equations, Sara Pollock, Wright State North Carolina State University, and Mile Krajcevski, University, and Leo Rebholz, Clemson University. University of South Florida. Research by Postdocs of the Alliance for Diversity in Women in Symplectic and Contact Geometry and To- Mathematics, Aloysius Helminck, University of Hawaii - pology, Bahar Acu, , Ziva Myer, Manoa, and Michael Young, Iowa State University. Duke University, and Yu Pan, Massachusetts Institute of Research from the Rocky Mountain-Great Plains Gradu- Technology (AMS-AWM). ate Research Workshop in Combinatorics, Michael Ferrara, University of Colorado Denver, Leslie Hogben, Iowa State University, Paul Horn, University of Denver, and Tyrrell Columbus, Ohio McAllister, University of Wyoming. Ohio State University Research in Mathematics by Early Career Graduate Students, Michael Bishop, Marat Markin, Khang Tran, and March 17–18, 2018 Oscar Vega, California State University, Fresno. Saturday – Sunday Research in Mathematics by Undergraduates and Stu- dents in Post-Baccalaureate Programs, Tamas Forgacs, Meeting #1136 CSU Fresno, Darren A. Narayan, Rochester Institute of Central Section Mark David Ward Technology, and , Purdue University Associate secretary: Georgia Benkart (AMS-MAA-SIAM). Announcement issue of Notices: December 2017 Set Theory, Logic and Ramsey Theory, Andrés Caicedo, Program first available on AMS website: January 31, 2018 Mathematical Reviews, and José Mijares, University of Program issue of electronic Notices: To be announced Colorado, Denver (AMS-ASL). Issue of Abstracts: Volume 39, Issue 2 Set-theoretic Topology (Dedicated to Jack Porter in Honor of 50 Years of Dedicated Research), Nathan Carlson, Deadlines California Lutheran University, Jila Niknejad, University For organizers: Expired of Kansas, and Lynne Yengulalp, University of Dayton. For abstracts: January 22, 2018 Special Functions and Combinatorics (in honor of Dennis Stanton’s 65th birthday), Susanna Fishel, Arizona State The scientific information listed below may be dated. University, Mourad Ismail, University of Central Florida, For the latest information, see www.ams.org/amsmtgs/ and Vic Reiner, University of Minnesota. sectional.html. Spectral Theory, Disorder and Quantum Physics, Ra- jinder Mavi and Jeffery Schenker, Michigan State Uni- Invited Addresses versity. Aaron Brown, University of Chicago, Title to be an- Stochastic Processes, Stochastic Optimization and Con- nounced. trol, Numerics and Applications, Hongwei Mei, University Tullia Dymarz, University of Wisconsin-Madison, Title of Central Florida, Zhixin Yang and Quan Yuan, Ball State to be announced. University, and Guangliang Zhao, GE Global Research. , Institute for Advanced Study, Title to be Strengthening Infrastructures to Increase Capacity announced. Around K-20 Mathematics, Brianna Donaldson, Ameri- can Institute of Mathematics, William Jaco and Michael Special Sessions Oehrtman, Oklahoma State University, and Levi Patrick, If you are volunteering to speak in a Special Session, you Oklahoma State Department of Education. should send your abstract as early as possible via the ab-

1356 Notices of the AMS Volume 64, Number 11 MEETINGS & CONFERENCES stract submission form found at www.ams.org/cgi-bin/ Homological Algebra (Code: SS 4A), Ela Celikbas and abstracts/abstract.pl. Olgur Celikbas, West Virginia University. (Code: SS 29A), Ernest Fontes, John Advances in Integral and Differential Equations (Code: E. Harper, Crichton Ogle, and Gabriel Valenzuela, Ohio SS 26A), Jeffrey T. Neugebauer, Eastern Kentucky Univer- State University. sity, and Min Wang, Rowan University. Lefschetz Properties (Code: SS 24A), Juan Migliore, Algebraic Coding Theory and Applications (Code: SS University of Notre Dame, and Uwe Nagel, University of 27A), Heide Gluesing-Luerssen, University of Kentucky, Kentucky. Christine A. Kelley, University of Nebraska-Lincoln, and Mathematical Modeling of Neuronal Networks (Code: Steve Szabo, Eastern Kentucky University. SS 36A), Janet Best, Ohio State University, Alicia Prieto Algebraic Combinatorics: Association Schemes, Finite Langarica, Youngstown State University, and Pamela B. Geometry, and Related Topics (Code: SS 15A), Sung Y. Pyzza, Ohio Wesleyan University. Song, Iowa State University, and Bangteng Xu, Eastern Multiplicative Ideal Theory and Factorization (in honor Kentucky University. of Tom Lucas retirement) (Code: SS 7A), Evan Houston, Algebraic Curves and Their Applications (Code: SS 17A), University of North Carolina, Charlotte, and Alan Loper, Artur Elezi, American University, Monika Polak, Maria Ohio State University. Curie-Sklodowska University (Poland) and University of Noncommutative Algebra and Noncommutative Al- Information Science and Technology (Mac, and Tony gebraic Geometry (Code: SS 16A), Jason Gaddis, Miami Shaska, Oakland University. University, and Robert Won, Wake Forest University. Algebraic and Combinatorial Aspects of Tropical Ge- Nonlinear Evolution Equations (Code: SS 9A), John ometry (Code: SS 11A), Maria Angelica Cueto, Ohio State Holmes and Feride Tiglay, Ohio State University. University, Yoav Len, University of Waterloo, and Martin Nonlinear Waves and Patterns (Code: SS 19A), Anna Ulirsch, University of Michigan. Ghazaryan, Miami University, Stephane Lafortune, Col- Algebraic, Combinatorial, and Quantum Invariants of lege of Charleston, and Vahagn Manukian and Alin Pogan, Miami University. Knots and Manifolds (Code: SS 6A), Cody Armond, Ohio Parameter Analysis and Estimation in Applied Dynami- State University, Mansfield, Micah Chrisman, Monmouth cal Systems (Code: SS 35A), Adriana Dawes, The Ohio University, and Heather Dye, McKendree University. State University, and Reginald L. McGee, Mathematical Analytical and Computational Advances in Mathemati- Biosciences Institute. cal Biology Across Scales (Code: SS 30A), Veronica Ciocanel Probabilistic and Extremal Graph Theory (Code: SS 32A), and Alexandria Volkening, Mathematical Biosciences Louis DeBiasio and Tao Jiang, Miami University. Institute. Probability in Convexity and Convexity in Probability Categorical, Homological and Combinatorial Methods in (Code: SS 2A), Elizabeth Meckes, Mark Meckes, and Elisa- Algebra (Celebrating the 80th birthday of S. K. Jain) (Code: beth Werner, Case Western Reserve University. SS 28A), Pedro A. Guil Asensio, University of Murcia, Ivo Quantum Symmetries (Code: SS 3A), David Penneys, Herzog, Ohio State University, Andre Leroy, University of The Ohio State University, and Julia Plavnik, Texas A & Artois, and Ashish K. Srivastava, Saint Louis University. M University. Coherent Structures in Interfacial Flows (Code: SS 14A), Recent Advances in Approximation Theory and Opera- Benjamin Akers and Jonah Reeger, Air Force Institute of tor Theory (Code: SS 1A), Jan Lang and Paul Nevai, The Technology. Ohio State University. Commutative and Combinatorial Algebra (Code: SS Recent Advances in Finite Element Methods for Partial 18A), Jennifer Biermann, Hobart and William Smith Col- Differential Equations (Code: SS 31A), Ching-shan Chou, leges, and Kuei-Nuan Lin, Penn State University, Greater Yukun Li, and Yulong Xing, The Ohio State University. Allegheny. Recent Advances in Packing (Code: SS 23A), Joseph Convex Bodies in Algebraic Geometry and Representa- W. Iverson, University of Maryland, John Jasper, South tion Theory (Code: SS 20A), Dave Anderson, Ohio State Dakota State University, and Dustin G. Mixon, The Ohio University, and Kiumars Kaveh, University of Pittsburgh. State University. Differential Equations and Applications (Code: SS 8A), Recent Development of Nonlinear Geometric PDEs (Code: King-Yeung Lam and Yuan Lou, Ohio State University, and SS 12A), Bo Guan, Ohio State University, Qun Li, Wright Qiliang Wu, Michigan State University. State University, Xiangwen Zhang, University of Califor- Function Spaces, Operator Theory, and Non-Linear nia, Irvine, and Fangyang Zheng, Ohio State University. Differential Operators (Code: SS 21A), David Cruz-Uribe, Several Complex Variables (Code: SS 13A), Liwei Chen, University of Alabama, and Osvaldo Mendez, University Kenneth Koenig, and Liz Vivas, Ohio State University. of Texas. Stochastic Analysis in Infinite Dimensions (Code: SS Geometric Methods in Shape Analysis (Code: SS 10A), Se- 22A), Parisa Fatheddin, Air Force Institute of Technology, bastian Kurtek and Tom Needham, Ohio State University. and Arnab Ganguly, Louisiana State University. Graph Theory (Code: SS 5A), John Maharry, Ohio State Structure and Representation Theory of Finite Groups University, Yue Zhao, University of Central Florida, and (Code: SS 33A), Justin Lynd, University of Louisiana at Xiangqian Zhou, Wright State University. Lafayette, and Hung Ngoc Nguyen, University of Akron.

December 2017 Notices of the AMS 1357 MEETINGS & CONFERENCES

Symmetry in Differential Geometry (Code: SS 34A), Sam- Vanderbilt University, and Michael Hull, University of uel Lin, Dartmouth College, Barry Minemyer, Bloomsburg Florida. University, and Ben Schmidt, Michigan State University. Commutative Algebra (Code: SS 8A), Florian Enescu The Mathematics of Phylogenetics (Code: SS 25A), Colby and Yongwei Yao, Georgia State University. Long, Mathematical Biosciences Institute. Difference Equations and Applications (Code: SS 2A), Topology and Geometry in Data Analysis (Code: SS Michael A. Radin, Rochester Institute of Technology, and 37A), Sanjeevi Krishnan and Facundo Memoli, Ohio State Youssef Raffoul, University of Dayton, Ohio. University. Evolution Equations and Applications (Code: SS 14A), Marcelo Disconzi, Chenyun Luo, Giusy Mazzone, and Gieri Simonett, Vanderbilt University. Nashville, Tennessee Function Spaces and Operator Theory (Code: SS 9A), Cheng Chu and Dechao Zheng, Vanderbilt University. Vanderbilt University Harmonic Analysis, Functional Analysis, and Their Ap- plications (Code: SS 11A), Akram Aldroubi and Keaton April 14–15, 2018 Hamm, Vanderbilt University, Michael Worthington, Geor- Saturday – Sunday gia Institute of Technology, and Alex Powell, Vanderbilt University. Meeting #1138 Hermitian Geometry (Code: SS 18A), Mehdi Lejmi, Bronx Southeastern Section Community College of CUNY, and Rares Rasdeaconu and Associate secretary: Brian D. Boe Ioana Suvaina, Vanderbilt University. Announcement issue of Notices: January 2018 Interactions between Geometry, Group Theory and Program first available on AMS website: February 22, 2018 Dynamics (Code: SS 13A), Jayadev Athreya, University Program issue of electronic Notices: To be announced of Washington, and Caglar Uyanik and Grace Work, Issue of Abstracts: Volume 39, Issue 2 Vanderbilt University. Deadlines Macdonald Polynomials and Related Structures (Code: SS 23A), Jennifer Morse, University of Virginia, and Dan For organizers: Expired Orr and Mark Shimozono, Virginia Polytechnic Institute For abstracts: February 13, 2018 and State University. Mathematical Chemistry (Code: SS 10A), Hua Wang, The scientific information listed below may be dated. Georgia Southern University. For the latest information, see www.ams.org/amsmtgs/ and Related Structures (Code: SS 5A), Carolyn sectional.html. Chun, United States Naval Academy, Deborah Chun and Invited Addresses Tyler Moss, West Virginia University Institute of Technol- ogy, and Jakayla Robbins, Vanderbilt University. Andrea Bertozzi, University of California Los Angeles, Partial Differential Equations and New Perspective of Title to be announced (Erdo˝s Memorial Lecture). J. M. Landsberg, Texas A & M University, Title to be Variational Methods (Code: SS 16A), Abbas Moameni, announced. Carleton University, Futoshi Takahashi, Osaka City Uni- Jennifer Morse, University of Virginia, Title to be an- versity, and Michinori Ishiwata, Osaka University. nounced. Probabilistic Models in Mathematical Physics (Code: SS Kirsten Wickelgren, Georgia Institute of Technology, 6A), Robert Buckingham, University of Cincinnati, Seung- Title to be announced. Yeop Lee, University of South Florida, and Karl Liechty, DePaul University. Special Sessions Quantization for Probability Distributions and Dynami- If you are volunteering to speak in a Special Session, you cal Systems (Code: SS 1A), Mrinal Kanti Roychowdhury, should send your abstract as early as possible via the ab- University of Texas Rio Grande Valley. stract submission form found at www.ams.org/cgi-bin/ Random Discrete Structures (Code: SS 22A), Lutz P abstracts/abstract.pl. Warnke, Georgia Institute of Technology, and Xavier Pérex-Giménez, University of Nebraska-Lincoln (AMS- Advances in Operator Algebras (Code: SS 7A), Scott AAAS). Atkinson, Dietmar Bisch, , and Jesse Pe- Recent Advances in Mathematical Biology (Code: SS terson, Vanderbilt University. 12A), Glenn Webb and Yixiang Wu, Vanderbilt University. Algebraic Geometry, Representation Theory, and Ap- Recent Advances on Complex Bio-systems and Their plications (Code: SS 21A), Shrawan Kumar, University of Applications (Code: SS 17A), Pengcheng Xiao, University North Carolina at Chapel Hill, J. M. Landsberg, Texas A&M of Evansville. University, and Luke Oeding, Auburn University. Recent Progress and New Directions in Homotopy Boundaries and Non-positive Curvature in Group Theory Theory (Code: SS 20A), Anna Marie Bohmann, Vanderbilt (Code: SS 15A), Spencer Dowdall and Matthew Hallmark, University, and Kirsten Wickelgren, Georgia Institute of Technology.

1358 Notices of the AMS Volume 64, Number 11 MEETINGS & CONFERENCES

Selected Topics in Graph Theory (Code: SS 3A), Songling of Technology, Mariela Carvacho, Universidad Tecnica Shan, Vanderbilt University, and David Chris Stephens Federico Santa Maria, Anthony Weaver, Bronx Commu- and Dong Ye, Middle Tennessee State University. nity College, the City University of New York, and Aaron Structural Graph Theory (Code: SS 4A), Joshua Fallon, Wootton, University of Portland. Louisiana State University, and Emily Marshall, Arcadia Biomathematics - Progress and Future Directions (Code: University. SS 4A), Hannah Callender Highlander, University of Port- Tensor Categories and Diagrammatic Methods (Code: SS land, Peter Hinow, University of Wisconsin - Milwaukee, 19A), Marcel Bischoff, Ohio University, and Henry Tucker, and Deena Schmidt, University of Nevada, Reno. University of California San Diego. Commutative Algebra (Code: SS 5A), Adam Boocher, University of Utah, and Irena Swanson, Reed College. Complex Analysis and Applications (Code: SS 11A), Portland, Oregon Malik Younsi, University of Hawaii Manoa. Portland State University Differential Geometry (Code: SS 19A), Christine Escher, Oregon State University, and Catherine Searle, Wichita April 14–15, 2018 State University. Saturday – Sunday Forest Modeling (Code: SS 20A), Gatziolis Demetrios, Pacific Northwest Research Station, US Forest Service, and Meeting #1137 Nikolay Strigul, Washington State University, Vancouver. Western Section General Relativity and Geometric Analysis (Code: SS Associate secretary: Michel L. Lapidus 13A), Paul T. Allen, Lewis & Clark College, Jeffrey Jaure- Announcement issue of Notices: January 2018 gui, Union College, and Iva Stavrov Allen, Lewis & Clark Program first available on AMS website: February 15, 2018 College. Program issue of electronic Notices: To be announced Geometric Measure Theory and Partial Differential Issue of Abstracts: Volume 39, Issue 2 Equations (Code: SS 12A), Mark Allen, Brigham Young University, and Spencer Becker-Kahn and Mariana Smit Deadlines Vega Garcia, University of Washington. For organizers: Expired Inverse Problems (Code: SS 2A), Hanna Makaruk, Los For abstracts: February 6, 2018 Alamos National Laboratory (LANL), and Robert Owcza- rek, University of New Mexico, Albuquerque & Los Alamos. The scientific information listed below may be dated. Mock Modular and Quantum Modular Forms (Code: SS For the latest information, see www.ams.org/amsmtgs/ sectional.html. 24A), Holly Swisher, Oregon State University, and Stepha- nie Treneer, Western Washington University. Invited Addresses Modeling, Analysis, and Simulation of PDEs with Multiple Sándor Kovács, University of Washington, Seattle, Title Scales, Interfaces, and Coupled Phenomena (Code: SS 17A), to be announced. Malgorzata Peszynska, Oregon State University. Elena Mantovan, California Institute of Technology, Moduli Spaces (Code: SS 21A), Renzo Cavalieri, Colo- Title to be announced. rado State University, and Damiano Fulghesu, Minnesota Dimitri Shlyakhtenko, University of California, Los State University Moorhead. Angeles, Title to be announced. Motivic homotopy theory (Code: SS 6A), Daniel Dugger, University of Oregon, and Kyle Ormsby, Reed College. Special Sessions Noncommutative Algebraic Geometry and Related Top- If you are volunteering to speak in a Special Session, you ics (Code: SS 16A), Jesse Levitt, University of Southern should send your abstract as early as possible via the ab- California, Hans Nordstrom, University of Portland, and stract submission form found at www.ams.org/cgi-bin/ Xinting Wang, Temple University. abstracts/abstract.pl. Nonsmooth Optimization and Applications(Dedicated to Prof. B. S. Mordukhovich on the occasion of his 70th birth- Algebraic Geometry and its Connections (Code: SS 9A), day) (Code: SS 7A), Mau Nam Nguyen, Portland State Uni- Sándor Kovács, University of Washington, Seattle, and versity, Hung M. Phan, University of Massachusetts Lowell, Karl Schwede, University of Utah, Salt Lake City. and Shawn Xianfu Wang, University of British Columbia. Algebraic Topology (Code: SS 23A), Angélica Osorno, Reed College, and Dev Sinha, University of Oregon. Numerical Methods for Partial Differential Equations Algebraic and Combinatorial Structures in Knot Theory (Code: SS 22A), Brittany A. Erickson and Jeffrey S. Ovall, (Code: SS 3A), Allison Henrich, Seattle University, Inga Portland State University. Johnson, Willamette University, and Sam Nelson, Clare- Pattern Formation in Crowds, Flocks, and Traffic (Code: mont McKenna College. SS 1A), J. J. P. Veerman, Portland State University, Alethea Automorphisms of Riemann Surfaces and Related Topics Barbaro, Case Western Reserve University, and Bassam (Code: SS 14A), S. Allen Broughton, Rose-Hulman Institute Bamieh, UC Santa Barbara.

December 2017 Notices of the AMS 1359 MEETINGS & CONFERENCES

Recent Advances in Actuarial Mathematics (Code: SS Algebraic Statistics (Code: SS 33A), Kaie Kubjas and 18A), Sooie-Hoe Loke, Central Washington University, and Elina Robeva, Massachusetts Institute of Technology. Enrique Thomann, Oregon State University. Algebraic, Geometric, and Topological Methods in Com- Spectral Theory (Code: SS 8A), Jake Fillman, Virginia binatorics (Code: SS 21A), Florian Frick, Cornell University, Tech, and Milivoje Lukic, Rice University. and Pablo Soberón, Northeastern University. Teaching and Learning in Undergraduate Mathematics Algorithmic Group Theory and Applications (Code: SS (Code: SS 15A), Natalie LF Hobson, Sonoma State Univer- 26A), Delaram Kahrobaei, City University of New York, sity, and Elise Lockwood, Oregon State University. and Antonio Tortora, University of Salerno. Wavelets, Frames, and Related Expansions (Code: SS Analysis and Geometry in Non-smooth Spaces (Code: 10A), Marcin Bownik, University of Oregon, and Darrin SS 5A), Nageswari Shanmugalingam and Gareth Speight, Speegle, Saint Louis University. University of Cincinnati. Arithmetic Dynamics (Code: SS 1A), Jacqueline M. Anderson, Bridgewater State University, Robert Bene- Boston, detto, Amherst College, and Joseph H. Silverman, Brown University. Arrangements of Hypersurfaces (Code: SS 2A), Graham Massachusetts Denham, University of Western Ontario, and Alexander I. Northeastern University Suciu, Northeastern University. Combinatorial Aspects of Nilpotent Orbits (Code: SS April 21–22, 2018 15A), Anthony Iarrobino, Northeastern University, Leila Saturday – Sunday Khatami, Union College, and Juliana Tymoczko, Smith College. Meeting #1139 Combinatorial Representation Theory (Code: SS 41A), Eastern Section Laura Colmenarejo, York University, Ricky Liu, North Associate secretary: Steven H. Weintraub Carolina State University, and Rosa Orellana, Dartmouth Announcement issue of Notices: January 2018 College. Program first available on AMS website: March 1, 2018 Connections Between Trisections of 4-manifolds and Program issue of electronic Notices: To be announced Low-dimensional Topology (Code: SS 32A), Jeffrey Meier, Issue of Abstracts: Volume 39, Issue 2 University of Georgia, and Juanita Pinzon-Caicedó, North Carolina State University. Deadlines Discretization in Geometry and Dynamics (Code: SS For organizers: Expired 36A), Richard Kenyon, Wai Yeung Lam, and Richard For abstracts: February 20, 2018 Schwartz, Brown University. Dynamical systems, Geometric Structures and Special The scientific information listed below may be dated. Functions (Code: SS 23A), Alessandro Arsie, University For the latest information, see www.ams.org/amsmtgs/ of Toledo, and Oksana Bihun, University of Colorado, sectional.html. Colorado Springs. Effective Behavior in Random Environments (Code: SS Invited Addresses 25A), Jessica Lin, McGill University, and Charles Smart, Jian Ding, University of Chicago, Title to be announced. University of Chicago. Edward Frenkel, University of California, Berkeley, Title Ergodic Theory and Dynamics in Combinatorial Number to be announced (Einstein Public Lecture in Mathematics). Theory (Code: SS 7A), Stanley Eigen and Daniel Glasscock, Valentino Tosatti, Northwestern University, Title to be Northeastern University, and Vidhu Prasad, University of announced. Massachusetts, Lowell. , École Polytechnique Fédérale de Extremal Graph Theory and Quantum Walks on Graphs Lausanne, Title to be announced. (Code: SS 13A), Sebastian Cioaba˘, University of Delaware, Mark Kempton, Harvard University, Gabor Lippner, Special Sessions Northeastern University, and Michael Tait, Carnegie Mel- If you are volunteering to speak in a Special Session, you lon University. should send your abstract as early as possible via the ab- Facets of Symplectic Geometry and Topology (Code: SS stract submission form found at www.ams.org/cgi-bin/ 3A), Tara Holm, Cornell University, Jo Nelson, Columbia abstracts/abstract.pl. University, and Jonathan Weitsman, Northeastern Uni- versity. Algebraic Number Theory (Code: SS 35A), Michael Bush, Geometries Defined by Differential Forms (Code: SS Washington and Lee University, Farshid Hajir, University 34A), Mahir Bilen Can, Tulane University, Sergey Grigo- of Massachusetts, and Christian Maire, Université Bour- rian, University of Texas Rio Grande Valley, and Sema gogne Franche-Comté. Salur, University of Rochester.

1360 Notices of the AMS Volume 64, Number 11 MEETINGS & CONFERENCES

Geometry and Analysis of Fluid Equations (Code: SS The Analysis of Dispersive Equations (Code: SS 39A), 28A), Robert McOwen and Peter Topalov, Northeastern Marius Beceanu, University at Albany, and Andrew Law- University. rie, Massachusetts Institute of Technology. Geometry of Moduli Spaces (Code: SS 10A), Ana-Marie The Gaussian Free Field and Random Geometry (Code: Castravet and Emanuele Macrí, Northeastern University, SS 12A), Jian Ding, University of Chicago, and Vadim Benjamin Schmidt, University of Texas, and Xiaolei Zhao, Gorin, Massachusetts Institute of Technology. Northeastern University. Topics in Qualitative Properties of Partial Differential Global Dynamics of Real Discrete Dynamical Systems Equations (Code: SS 27A), Changfeng Gui, University of (Code: SS 30A), M. R. S. Kulenovic´ and O. Merino, Univer- Texas at San Antonio, Changyou Wang, Purdue University, sity of Rhode Island. and Jiuyi Zhu, Louisiana State University. Harmonic Analysis and Partial Differential Equations (Code: SS 29A), Donatella Danielli, Purdue University, and Topics in Toric Geometry (Code: SS 17A), Ivan Martino, Irina Mitrea, Temple University. Northeastern University, and Emanuele Ventura, Texas Homological Commutative Algebra (Code: SS 11A), Sean A&M University. Sather-Wagstaff, Clemson University, and Oana Veliche, Topology of Biopolymers (Code: SS 18A), Erica Flapan, Northeastern University. Pomona College, and Helen Wong, Carleton College. Hopf Algebras, Tensor Categories, and Homological Algebra (Code: SS 8A), Cris Negron, Massachusetts Insti- tute of Technology, Julia Plavnik, Texas A&M, and Sarah Shanghai, People’s Witherspoon, Texas A&M University. Mathematical Perspectives in Quantum Information Theory (Code: SS 24A), Aram Harrow, Massachusetts In- Republic of China stitute of Technology, and Christopher King, Northeastern Fudan University University. Mathematical Problems of Relativistic Physics: Classical June 11–14, 2018 and Quantum (Code: SS 37A), Michael Kiessling and A. Monday – Thursday Shadi Tahvildar-Zadeh, Rutgers University. Modeling of Biological Processes (Code: SS 38A), Simone Meeting #1140 Cassani and Sarah Olson, Worcester Polytechnic Institute. New Developments in Inverse Problems and Imaging Associate secretary: Steven H. Weintraub (Code: SS 9A), Ru-Yu Lai, University of Minnesota, and Announcement issue of Notices: April 2018 Ting Zhou, Northeastern University. Program first available on AMS website: Not applicable Noncommutative Algebra and Representation Theory Program issue of electronic Notices: Not applicable (Code: SS 22A), Van C. Nguyen, Hood College, and Alex Issue of Abstracts: Not applicable Martsinkovsky and Gordana Todorov, Northeastern University. Deadlines Nonlinear Reaction-Diffusion Equations and Their Appli- For organizers: To be announced cations (Code: SS 31A), Nsoki Mavinga and Quinn Morris, For abstracts: To be announced Swarthmore College. Nonlinear and Stochastic Partial Differential Equations The scientific information listed below may be dated. and Applications (Code: SS 19A), Nathan Glatt-Holtz and For the latest information, see www.ams.org/amsmtgs/ Vincent Martinez, Tulane University, and Cecilia Mon- internmtgs.html. daini, Texas A&M University. Numerical Methods and Applications (Code: SS 16A), Invited Addresses Vera Babenko, Ithaca College. Yu-Hong Dai, Academy of Mathematics and System Optimization Under Uncertainty (Code: SS 40A), Ying- Sciences, Title to be announced. dong Lu and Mark S. Squillante, IBM Research. Kenneth A. Ribet, University of California, Berkeley, Polytopes and Discrete Geometry (Code: SS 6A), Gabriel Cunningham, University of Massachusetts, Boston, Mark Title to be announced. Mixer, Wentworth Institute of Technology, and Egon Richard M. Schoen, University of California, Irvine, Title Schulte, Northeastern University. to be announced. Regularity of PDEs on Rough Domains (Code: SS 14A), , University of Michigan, Title to be announced. Murat Akman, University of Connecticut, and Max Engel- Chenyang Xu, Peking University, Title to be announced. stein, Massachusetts Institute of Technology. Jiangong You, Nankai University, Title to be announced. Relations Between the History and Pedagogy of Math- ematics (Code: SS 20A), Amy Ackerberg-Hastings, and David L. Roberts, Prince George’s Community College. Singularities of Spaces and Maps (Code: SS 4A), Terence Gaffney and David Massey, Northeastern University.

December 2017 Notices of the AMS 1361 MEETINGS & CONFERENCES

Invited Addresses Newark, Delaware Mihalis Dafermos, Princeton University, Title to be announced. University of Delaware Jonathan Hauenstein, University of Notre Dame, Title to be announced. September 29–30, 2018 Kathryn Mann, University of California Berkeley, Title Saturday – Sunday to be announced. Meeting #1141 Eastern Section Ann Arbor, Michigan Associate secretary: Steven H. Weintraub Announcement issue of Notices: June 2018 University of Michigan, Ann Arbor Program first available on AMS website: August 9, 2018 Program issue of electronic Notices: To be announced October 20–21, 2018 Issue of Abstracts: Volume 39, Issue 3 Saturday – Sunday

Deadlines Meeting #1143 For organizers: February 28, 2018 Central Section For abstracts: July 31, 2018 Associate secretary: Georgia Benkart Announcement issue of Notices: July 2018 Program first available on AMS website: August 30, 2018 The scientific information listed below may be dated. Program issue of electronic Notices: To be announced For the latest information, see www.ams.org/amsmtgs/ Issue of Abstracts: Volume 39, Issue 4 sectional.html. Deadlines Invited Addresses For organizers: March 20, 2018 Leslie Greengard, New York University, Title to be an- For abstracts: August 21, 2018 nounced. Elisenda Grigsby, Boston College, Title to be announced. The scientific information listed below may be dated. Davesh Maulik, Massachusetts Institute of Technology, For the latest information, see www.ams.org/amsmtgs/ Title to be announced. sectional.html.

Invited Addresses Fayetteville, Elena Fuchs, University of Illinois Urbana-Champaign, Title to be announced. Andrew Putman, University of Notre Dame, Title to be Arkansas announced. University of Arkansas Charles Smart, University of Chicago, Title to be an- nounced. October 6–7, 2018 Special Sessions Saturday – Sunday If you are volunteering to speak in a Special Session, you Meeting #1142 should send your abstract as early as possible via the ab- Southeastern Section stract submission form found at www.ams.org/cgi-bin/ abstracts/abstract.pl. Associate secretary: Brian D. Boe Announcement issue of Notices: July 2018 From Hyperelliptic to Superelliptic Curves (Code: SS 6A), Program first available on AMS website: August 16, 2018 Tony Shaska, Oakland University, Nicola Tarasca, Rutgers Program issue of electronic Notices: To be announced University, and Yuri Zarhin, Pennsylvania State University. Issue of Abstracts: Volume 39, Issue 3 Geometry of Submanifolds, in Honor of Bang-Yen Chens 75th Birthday (Code: SS 1A), Alfonso Carriazo, University Deadlines of Sevilla, Ivko Dimitric, Penn State Fayette, Yun Myung For organizers: March 6, 2018 Oh, Andrews University, Bogdan D. Suceava, California For abstracts: August 7, 2018 State University, Fullerton, Joeri Van der Veken, Univer- sity of Leuven, and Luc Vrancken, Universite de Valen- The scientific information listed below may be dated. ciennes. For the latest information, see www.ams.org/amsmtgs/ Interactions between Algebra, Machine Learning and sectional.html. Data Privacy (Code: SS 3A), Jonathan Gryak, University of

1362 Notices of the AMS Volume 64, Number 11 MEETINGS & CONFERENCES

Michigan, Kelsey Horan, CUNY Graduate Center, Delaram Coupling in Probability and Related Fields (Code: SS 3A), Kahrobaei, CUNY Graduate Center and New York Univer- Sayan Banerjee, University of North Carolina, Chapel Hill, sity, Kayvan Najarian and Reza Soroushmehr, University and Terry Soo, University of Kansas. of Michigan, and Alexander Wood, CUNY Graduate Center. Homological Aspects of Noncommutative Algebra and Random Matrix Theory Beyond Wigner and Wishart Geometry (Code: SS 2A), Dan Rogalski, University of Cali- (Code: SS 2A), Elizabeth Meckes and Mark Meckes, Case fornia San Diego, Sarah Witherspoon, Texas A&M Univer- Western Reserve University, and Mark Rudelson, Univer- sity, and James Zhang, University of Washington, Seattle. sity of Michigan. Mathematical Biology with a focus on Modeling, Analy- Self-similarity and Long-range Dependence in Stochastic sis, and Simulation (Code: SS 1A), Jim Cushing, The Univer- Processes (Code: SS 4A), Takashi Owada, Purdue Univer- sity of Arizona, Saber Elaydi, Trinity University, Suzanne sity, Yi Shen, University of Waterloo, and Yizao Wang, Sindi, University of California, Merced, and Abdul-Aziz University of Cincinnati. Yakubu, Howard University. Structured Homotopy Theory (Code: SS 5A), Thomas Fiore, University of Michigan, Dearborn, Po Hu and Dan Isaksen, Wayne State University, and Igor Kriz, University Baltimore, Maryland of Michigan. Baltimore Convention Center, Hilton Baltimore, and Baltimore Marriott Inner San Francisco, Harbor Hotel California January 16–19, 2019 Wednesday – Saturday San Francisco State University Meeting #1145 October 27–28, 2018 Joint Mathematics Meetings, including the 125th Annual Saturday – Sunday Meeting of the AMS, 102nd Annual Meeting of the Math- ematical Association of America (MAA), annual meetings Meeting #1144 of the Association for Women in Mathematics (AWM)and Western Section the National Association of Mathematicians (NAM), and the Associate secretary: Michel L. Lapidus winter meeting of the Association of Symbolic Logic (ASL), Announcement issue of Notices: July 2018 with sessions contributed by the Society for Industrial and Program first available on AMS website: September 6, 2018 Applied Mathematics (SIAM). Program issue of electronic Notices: To be announced Associate secretary: Steven H. Weintraub Issue of Abstracts: Volume 39, Issue 4 Announcement issue of Notices: October 2018 Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: March 27, 2018 Issue of Abstracts: To be announced For abstracts: August 28, 2018 Deadlines The scientific information listed below may be dated. For organizers: April 2, 2018 For the latest information, see www.ams.org/amsmtgs/ For abstracts: To be announced sectional.html.

Invited Addresses Srikanth B. Iyengar, University of Utah, Title to be an- nounced. Sarah Witherspoon, Texas A&M University, Title to be announced. Abdul-Aziz Yakubu, Howard University, Title to be announced.

Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the ab- stract submission form found at www.ams.org/cgi-bin/ abstracts/abstract.pl.

December 2017 Notices of the AMS 1363 MEETINGS & CONFERENCES Auburn, Alabama Quy Nhon City, Auburn University Vietnam March 15–17, 2019 Quy Nhon University Friday – Sunday June 10–13, 2019 Meeting #1146 Monday – Thursday Southeastern Section Associate secretary: Brain D. Boe Associate secretary: Brian D. Boe Announcement issue of Notices: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Issue of Abstracts: To be announced Deadlines Deadlines For organizers: To be announced For organizers: To be announced For abstracts: To be announced For abstracts: To be announced Honolulu, Hawaii Binghamton, New University of Hawaii at Manoa York

March 22–24, 2019 Binghamton University Friday – Sunday October 12–13, 2019 Meeting #1147 Saturday – Sunday Eastern Section Central Section Associate secretary: Steven H. Weintraub Associate secretaries: Georgia Benkart and Michel L. Announcement issue of Notices: To be announced Lapidus Program first available on AMS website: To be announced Announcement issue of Notices: To be announced Program issue of electronic Notices: To be announced Program first available on AMS website: To be announced Issue of Abstracts: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Deadlines For organizers: March 12, 2019 Deadlines For abstracts: To be announced For organizers: May 15, 2018 For abstracts: January 22, 2019

The scientific information listed below may be dated. Gainesville, Florida For the latest information, see www.ams.org/amsmtgs/ University of Florida sectional.html. November 2–3, 2019 Invited Addresses Saturday – Sunday , Harvard University, Title to be announced Southeastern Section (Einstein Public Lecture in Mathematics). Associate secretary: Brian D. Boe Aaron Naber, Northwestern University, Title to be an- Announcement issue of Notices: To be announced nounced. Program first available on AMS website: To be announced Deanna Needell, University of California, Los Angeles, Program issue of electronic Notices: To be announced Title to be announced. Issue of Abstracts: To be announced Katherine Stange, University of Colorado, Boulder, Title to be announced. Deadlines Andrew Suk, University of Illinois at Chicago, Title to For organizers: To be announced be announced. For abstracts: To be announced

1364 Notices of the AMS Volume 64, Number 11 MEETINGS & CONFERENCES Denver, Colorado Moving? Colorado Convention Center

January 15–18, 2020 Please make sure that the AMS Notices and Bulletin find their new home. Wednesday – Saturday Joint Mathematics Meetings, including the 126th Annual

ISSN 0002-9920 (print) Meeting of the AMS, 103rd Annual Meeting of the Math- ISSN 1088-9477 (online) ematical Association of America (MAA), annual meetings Volume 61, Number 1 of the American Mathematical Society

January 2014

The Legacy of Vladimir of the Association for Women in Mathematics (AWM) and Andreevich Steklov page 9

Common Core State Standards for Geometry: An Alternative Approach page 24 the National Association of Mathematicians (NAM), and the Infrastructure: Structure Inside the Class Group of a Real Quadratic Field page 36

Hypergeometric Functions, How Special Are winter meeting of the Association of Symbolic Logic (ASL), They? page 48

Knoxville Meeting page 100

Baltimore Meeting with sessions contributed by the Society for Industrial and page 103 Albuquerque Meeting page 107

Lubbock Meeting page 111

Tel Aviv Meeting Applied Mathematics (SIAM) page 114

About the cover: Plain geometry (see page 88) Associate secretary: Michel L. Lapidus Announcement issue of Notices: October 2019 Program first available on AMS website: November 1, 2019 • Program issue of electronic Notices: To be announced Email your new address to us: Issue of Abstracts: To be announced [email protected] • or make the change yourself online at: Deadlines www.ams.org/cml-update For organizers: April 1, 2019 • or send the information to: For abstracts: To be announced Member and Customer Services American Mathematical Society 201 Charles Street Washington, District Providence, RI 02904-2294 USA Phone: (800) 321-4267 (US & Canada) of Columbia (401) 455-4000 (Worldwide) Walter E. Washington Convention Center

January 6–9, 2021 Wednesday – Saturday Joint Mathematics Meetings, including the 127th Annual Meeting of the AMS, 104th Annual Meeting of the Math- ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the winter meeting of the Association of Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Associate secretary: Brian D. Boe Announcement issue of Notices: October 2020 Program first available on AMS website: November 1, 2020 Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced

Deadlines For organizers: April 1, 2020 For abstracts: To be announced

December 2017 Notices of the AMS 1365 ISSN 0002-9920 (print) ISSN 1088-9477 (online)

of the American Mathematical Society June/July 2017 Volume 64, Number 6

Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance page 543 The Open Mathematics of Crystallization page 551

Topological Phase Transitions ISSN 0002-9920 (print) ISSN 1088-9477 (online) and Topological Phases of Matter page 557 Denton Meeting 2017 Notices Issue Index page 646 Buffalo Meeting of the Americanpage 652 Mathematical Society October 2017 Orlando Meeting Volume 64, Number 9 page 657

The Rigidity of Frameworks: Theory and Applications page 973 Search for specifi c content that appeared in the 2017 Notices. Dido’s Problem and Its Impact on Modern Mathematics page 980 San Diego Meeting—JMM 2018 page 1067

Cover artwork from Jessica Striker's feature on algebraic combinatorics This year-end Index is presented below in two easily searchable Parts: ISSN 0002-9920 (print) (see pp. 538 and 543). ISSN 1088-9477 (online) Part I: Issue at-a-Glance, is organized by specifi c Volume and Issue . This Part provides of the American Mathematical Society April 2017 Volume 64, Number 4

AMS Prize Announcements page 311 readers with a quick snapshot of the main articles featured in each specifi c Notices issue . Spring Sectional Sampler

page 333

AWM Research Symposium 2017 Lecture Sampler page 341

Mathematics and Statistics Part II: Societal Record, is organized alphabetically under each category heading . This Part Awareness Month page 362 provides readers with a searchable listing of all content of record for the Society, including: elections, awards, meetings, news, opportunities, annual AMS reports, surveys, grants,

About the Cover: How Minimal Surfaces Converge to a Foliation (see page 307) fellowships, etc . Issue at-a-Glance

January 2017 DEATH NOTICES/MEMORIAMS Volume 64, Issue I • pages 1–96 Ablow, Clarence M .; Anastasio, Salvatore; Brungs, Hans Heinrich; Cooper, Mark Alan; Craig, William; Enneking, Marj; Helmke, FEATURES Uwe R .; Huber-Dyson, Verena; Jakobsen, James F .; Kronheimer, E . H .; Moore, John C .; Pedersen, Jean J .; Reid, William H ., Satake, Hodge Theory of Matroids (Karim Adiprasito, June Huh, Eric Katz) Ichiro; Sexauer, Norman E .; Shapley, Lloyd S .; Silverman, Robert; ...... 26 Sinke, Carl J .; Smedinghoff, Scott ...... 62 JMM 2017 Lecture Sampler (Barry Simon, Alice Silverberg, Pelczynski, Aleksander (Olek) (1932–2012) ...... 54 Lisa Jeffrey, , Anna Wienhard, Donald St . P . Richards, Tobias Holck Colding, Wilfrid Gangbo, Ingrid February 2017 Daubechies) ...... 8 Volume 64, Issue 2 • pages 97–192 COMMUNICATIONS 2017 Annual Meeting of the American Association for the FEATURES Advancement of Science (Andy Magid) ...... 73 (A) Conceptual Breakthrough in (Henry Cohn) 102 Aleksander (Olek) Pelczynski (1932–2012) (Edited by Joe Diestel) (The) Partitionability Conjecture (Art M . Duval, Caroline J . Klivans, and Jeremy L . Martin) ...... 117 ...... 54 AMS Waldemar J . Trjitzinsky Awards Program Celebrates COMMUNICATIONS Its Twenty-Fifty Year (Allyn Jackson, AMS Trjitzinsky Fund Mathematical Expression of a Global Environmental Catastrophe announcement) ...... 44 (Daniel H . Rothman) ...... 138 My Year as an AMS Congressional Fellow (Anthony J . Macula) 40 Report on Bridges 2016—In Memory of Our Founder, Reza Sarhangi (1952–2016) (Carlo H . Séquin) ...... 152 GRADUATE STUDENT SECTION What Does Active Learning Mean For Mathematicians? (Benjamin Blog ...... 38 Braun, Priscilla Bremser, Art M . Duval, Elise Lockwood, Diana Interview with Arthur Benjamin (Alexander Diaz-Lopez) ...... 32 White) ...... 124 WHAT IS…a Complex Symmetric Operator? (Stephan Ramon GRADUATE STUDENT SECTION Garcia, communicated by Steven J . Miller and Cesar E . Silva) . 35 Blog ...... 135 REVIEWS Interview with Thomas Grandine (Alexander Diaz-Lopez) . . . . . 130 Origins of Mathematical Words (Andrew I . Dale) ...... 47 WHAT IS…Benford’s Law? (Arno Berger, Theodore P . Hill, Communicated by Cesar E . Silva) ...... 132 FROM THE EDITORS DOCEAMUS Diversity in Mathematics ...... 4 (The) Fulbright Specialist Program (Noureen A . Khan) ...... 158 BOOKSHELF ...... 50 THE BACK PAGE ...... 96

1366 Notices of the Ams Volume 64, Number 11 2017 Index

REVIEWS April 2017 (The) Mathematics of Various Entertaining Subjects (Brian Hayes) . . Volume 64, Issue 4 • pages 305–432 ...... 163 BOOKSHELF ...... 162 FEATURES THE BACK PAGE ...... 192 2017 AWM Lecture Sampler (Ruth Charney, Kristen Lauter, Svitlana Mayboroda, Linda Petzold, Ami Radunskaya) . . . . . 341 DEATH NOTICES/MEMORIAMS AMS Spring Sectional Sampler (Michael Hitrick, Sarah C . Koch, Gauchman, Hillel (1937–2016) ...... 150 Andrew Raich, Daniel Rogalski) ...... 333 March 2017 (A) New Golden Age of Minimal Surfaces (Joaquín Pérez) . . 347. Volume 64, Issue 4 • pages 193–304 COMMUNICATIONS IMAGINARY—a How-to Guide for Math Exhibitions (Andreas FEATURES Daniel Matt) ...... 368 Ad Honorem Sir Andrew J . Wiles, An Introduction (Christopher Strengthening Doctoral Programs in Mathematics Education: A Skinner) ...... 197 Continuous Process (Barbara Reys, Robert Reys) ...... 386 's Marvelous Proof (Henri Darmon) ...... 209 (The) World War II Origins of Mathematics Awareness (Michael J . Interview with Abel Laureate Sir Andrew J . Wiles (Martin Raussen, Barany) ...... 363 Christian Skau) ...... 198 GRADUATE STUDENT SECTION (The) Mathematical Works of Andrew Wiles (Christopher Blog ...... 431. Skinner, with contributions from Karl Rubin, Barry Mazur, Mirela Interview with (Alexander Diaz-Lopez) ...... Çiperiani, Chandrashekhar Kahre, Jack Thorne) ...... 217 Underrepresented Students in Topology and Algebra Research COMMUNICATIONS Symposium (USTARS) ...... 383. 100 Years Ago: Joan Clarke (Allyn Jackson, Ralph Erskine, I J. . WHAT IS…Tropical Geometry? (Eric Katz, Communicated by (Jack) Good, Eric A . Weiss) ...... 252 Cesar E . Silva) ...... Commutative Algebra Provides a Big Surprise for Craig Huneke's ...... 380 Birthday (Irena Swanson, Communicated by Tom Garrity) . . 256. OPINION Interview with New AMS President Kenneth A . Ribet (Allyn (A) Great Time to Be a Statistician (Ronald L . Wasserstein) . . .377 Jackson) ...... 229. (A) National Mathematics Festival and a Movement (David Math in Brazil: Sowing the Seeds (Marcelo Viana) ...... 234 Eisenbud and Kirsten Bohl) ...... 375 Math PhD Careers: New Opportunities Emerging Amidst Crisis BOOKSHELF ...... 407 (Yuliy Baryshnikov, Lee DeVille, Richard Laugesen) . . . . . 260 ABOUT THE COVER ...... 307 GRADUATE STUDENT SECTION THE BACK PAGE ...... 432 Interview with Ryan Haskett (Alexander Diaz-Lopez) . . . . . 239 DEATH NOTICES/MEMORIAMS WHAT IS…an Elliptic Curve? (Harris B . Daniels, Álvaro Lozano-

Robledo, Communicated by Steven J . Miller and Cesar E . Silva) . . Fuglister, Frederick J ...... 399 241 May 2017 OPINION Volume 64, Issue 5 • pages 433–536 (The) Common Vision Project: Four Reactions (William Yslas Vélez, Amy Cohen, Priscilla Bremser, Douglas Mupasiri, FEATURES Communicated by Harriet Pollatsek) ...... 245. AMS Spring Eastern Sectional Sampler (Fernanado Codá Marques, REVIEWS James Maynard, Kavita Ramanan) ...... 437. (A) Doubter’s Almanac (Sheldon Axler) ...... 265. MCA Invited Speakers Lecture Sampler (Andrew Granville ,Yuval THE BACK PAGE ...... 304. Peres) ...... 443 DEATH NOTICES/MEMORIAMS COMMUNICATIONS W . Wistar Comfort (1933–2016) (Martha Wistar Comfort) . . .275 2017 Breakthrough Prizes in Mathematics and Fundamental Physics Awarded ...... 498 Robert Seeley (1932–2016) (Ethan Bolker) ...... 276

December 2017 Notices of the AMS 1367 2017 Index

Big is Beautiful: Illinois Wins Exemplary Program Award (Allyn BOOKSHELF ...... 642 Jackson) ...... 481 COMMENTARY CeMEAI: The Brazilian Center and Its Mathematics Research for Industry (E . G . Birgin, J . A . Cuminato, J . M . Martínez, T . Pereira) . LETTERS TO THE EDITOR ...... 450 Survey on Math Post-docs (Amy Cohen) ...... 541. Fefferman and Schoen Awarded 2017 Wolf Prize in Mathematics Women's History Month: Don't Put a Man on the Cover (Autumn ...... 496 Kent) ...... 540. (The) Mathematics Genealogy Project Comes of Age at Twenty- THE BACK PAGE ...... 672 one (Colm Mulcahy, Communicated by Steven J . Miller) . . 466 DEATH NOTICES/MEMORIAMS GRADUATE STUDENT SECTION Cho, Choong Yun; Davis, Ruth M ;. Enneking, Marj; Gackstatter, Interview with Reinhard Laubenbacher (Alexander Diaz-Lopez) . . . Fritz; Gauchman, Hillel; Huber-Dyson, Verena; Karle, Jerome; ...... 456 Marcus, Solomon A ;. McCartin, Brian J .; Myung, Hyo Chul; Newell, David C ;. Philip, A . G . Davis; Shapley, Lloyd S ,. Sheffield, WHAT IS…a Virtual Knot? (Daniel S . Silver, Susan G . Williams, Roy D ,. Shenk, Al; Silverman, Robert; Wisner, Robert J . . . . 627. Communicated by Cesar E . Silva) ...... 461 (1934–2017) ...... 604 OPINION Can Mathematics Light the Way? (Allyn Jackson) ...... 448 August 2017 REVIEWS Volume 64, Issue 7 • pages 673–808 Math Graphics: A Review of Two Books (Judith Roitman) . . . 504 FEATURES BOOKSHELF ...... 508 (The) Mathematics of Gravitational Waves . A Two-Part Feature THE BACK PAGE ...... 536 ...... 684 Introduction (Christina Sormani) ...... 685 June/July 2017 Part One: How the Green Light Was Given for Gravitational Volume 64, Issue 6 • pages 537–672 Wave Search (C . Denson Hill, Paweł Nurowski ...... 686 FEATURES Part Two: Gravitational Waves and Their Mathematics (Lydia Bieri, David Garfinkle, and Nicolás Yunes ...... 693 Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance (Jessica Striker) COMMUNICATIONS (Communicated by Cesar E . Silva) ...... 543 Ellenberg in Move Gifted (Allyn Jackson) ...... 779. (The) Open Mathematics of Crystallization (Charles Radin) . . .551 (The) Global Math Project: Uplifting Mathematics for All (James Scientific Background on the Nobel Prize in Physics 2016: Tanton, Brianna Donaldson) ...... 712 Topological Phase Transitions and Topological Phases of Michèle Audin, Mathematician and Writer (Allyn Jackson) . . . 761 Matter (Compiled by the Class for Physics of the Royal Swedish (The) Travel Ban: Affected Mathematicians Tell Their Stories Academy of Sciences) ...... 557 (Alexander Diaz-Lopez, Allyn Jackson, and Stephen Kennedy) . . COMMUNICATIONS ...... 678 (The) “Wide Influence” of Leonard Eugene Dickson (Della (A) Conversation with Mu-Fa Chen ...... 616. Dumbaugh, Amy Shell-Gellasch) ...... 772 (A) Vision for CBMS ...... 590 GRADUATE STUDENT SECTION Joseph B . Keller (1923–2016) ...... 606 Interview with Karen E . Smith (Laure Flapan, Communicated by ...... 592 Awarded Abel Prize Alexander Diaz-Lopez) ...... 718 GRADUATE STUDENT SECTION WHAT IS…a CR Submanifold? (Phillip S . Harrington, Andrew Blog ...... 582 Raich, Communicated by Cesar E . Silva) ...... 722 Interview with Lucas Sabalka (Melinda Lanius) ...... 576 OPINION WHAT IS…a Generalised Mean-Curvature Flow? (Hui Yu, International Mobility and US Mathematics (Moon Duchin) . . 682 Communicated by Cesar E . Silva) ...... 580 Post-Quantum —A New Opportunity and Challenge DOCEAMUS for the Mathematics Community (Jintai Ding, Daniel Smith- Tone) ...... 709 What Is Inquiry-Based Learning? ...... 570 REVIEWS FILM REVIEW Gifted (Frank Morgan) ...... 716. Hidden Figures (Fern Y . Hunt) ...... 620

1368 Notices of the AMS Volume 64, Number 11 2017 Index

One Hundred Twenty-One Days (John McCleary) ...... 758 COMMENTARY LETTERS TO THE EDITOR BOOKSHELF ...... 778 Hayman's List (Eleanor Lingham) ...... 814 COMMENTARY In Support of Piper Harron (Nancy C . Wrinkle) ...... 814 LETTERS TO THE EDITOR I Never Met , but I Knew Her So Well (Mariel (The) Daughters of John Adams (Timothy Chow) ...... 676 Vazquez) ...... 813 Online Survey of Publishing Issues (Mark C . Wilson) ...... 676 ERRATUM ...... 815 THE BACK PAGE ...... 808. THE BACK PAGE ...... 968. DEATH NOTICES/MEMORIAMS THE AMS Komaravolu S . Chandrasekharan (1920–2017) ...... 771. FROM THE SECRETARY'S OFFICE September 2017 2017 American Mathematical Society Elections Volume 64, Issue 8 • pages 809–968 Table of Contents ...... 837 FEATURES List of Candidates ...... 838 AMS Fall Sectional Sampler (Christine Heitsch, Jonathan R . Election Information ...... 838 Kujawa, Govind Menon, Kevin M . Pilgrim, Bruce Sagan) . . . 816 Proposed Amendments to the Bylaws ...... 840 Two Optimization Problems in Thermal Insulation (Dorin Bucur, Nominations for President ...... 842 Giuseppe Buttazzo, Carlo Nitsch ...... 830 Biographies of Candidates 2017 ...... 845 COMMUNICATIONS Current Bylaws ...... 860 Kollár and Voisin Awarded Shaw Prize ...... 922 Call for Suggestions ...... 865 Mary Fairfax Somerville, Queen of Science (Elisabetta Strickland) ...... 929 Nominations by Petition ...... 866 Mathematics Applied to the Study of Bicycles (Jim Papadopoulous) ...... 888 October 2017 Volume 64, Issue 8 • pages 969–1128 My Life (Dusa McDuff) ...... 892 GRADUATE STUDENT SECTION FEATURES Making Matrices Better: Geometry and Topology of Singular Value Dido's Problem and Its Impact on Modern Mathematics and Polar Decomposition (Dennis DeTurck, Amora Elsaify, (Catherine Bandle) ...... 980 Herman Gluck, Benjamin Grossmann, Joseph Ansel Hoisington, Anusha M . Krishnan, Jianru Zhang) ...... 876. (The) Rigidity of Frameworks: Theory and Applications (Jessica Sidman, Audrey St . John) ...... 973 Blog ...... Interview with Kelsey Houston-Edwards (Alexander Diaz-Lopez) . . COMMUNICATIONS 870 Euphemia Lofton Haynes: Bringing Eduction Closer to the “Goal of WHAT IS…a Sobolev Orthogonal Polynomial? (Francisco Marcel- Perfection” (Susan E . Kelly, Carly Shinners, Katherine Zoroufy) . . lán, Juan J . Moreno-Balcázar, Communicated by Cesar E . Silva) ...... 995 873 GRADUATE STUDENT SECTION OPINION Gerrymandering, Sandwiches, and Topology (Pablo Soberón, Communicated by Cesar E . Silva) ...... 1010 Advocating for Diversity and Inclusion in Faculty Hiring (Darryl Yong, Sumun Pendakur) ...... 897 Blog ...... Beating the Lecture-Textbook Trap with Active Learning and Interview with Evelyn Lamb (Alexander Diaz-Lopez) . . . . . 1005 Rewards for All (David Pengelley) ...... 903 WHAT IS…Perimeter? (Editors) ...... 1009 REVIEWS OPINION Weapons of Math Destruction (Mary Poovey) ...... 933 Ensuring That Mathematics is Relevant in a World of Data Science (Johanna S . Hardin, Nicholas J . Horton (Communicated by ...... 936 BOOKSHELF Benjaion Braun) ...... 986 ABOUT THE COVER ...... 811 (A) Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap (Mira Bernstein, Moon Duchin) ...... 1020

December 2017 Notices of the AMS 1369 2017 Index

Flexibility in the Mathematics Major Would Benefit Students and COMMENTARY Society (William Yslas Vélez) ...... 992 LETTERS TO THE EDITOR REVIEWS Make March 20th Hypatia Day (Ari Belenkiy) ...... 1134 (The) Mathematician's Shiva (Dennis DeTurck) ...... 1043 More to Internationalism than Migration (Michael J . Barany) . 1133 BOOKSHELF ...... 1046 Presidential Nomination by Petition? (Joe Auslander) . . . . .1133 THE BACK PAGE ...... 1047 Travel Bank, Lincoln, and Beethoven (Lyla Fadali) ...... 1133 THE AMS We Don't Need “so-called” (Jim Stasheff) ...... 1134. JMM 2017 Photo Key and Montage ...... 359, 360 THE BACK PAGE ...... 1196. JMM 2018 Employment Center ...... 1041 DEATH NOTICES/MEMORIAMS JMM 2018 Program Announcements ...... 1067 Apostol, Tom M ;. Barth, Karl F .; Collins, Heron S .; Douglas, Jim Jr ;. JMM 2018 City Map ...... 1096 Feferman, Solomon; Gani, Joseph M .; Golomb, Solomon W .; Graves, JMM 2018 Meeting Registration Forms ...... 1127, 1128 William H ;. Kalman, Rudolf; Kenney, Margaret J .; Mimica, Ante; Neveu, Jacques; Otermat, Scott C .; Otway, Thomas H ;. Reade, JMM 2018 Program at a Glance ...... 1113 Maxwell O .; Reid, Walter M ;. Robinson, Ivor; Rogers, James T . Jr ;. Shaw, Ronald; Stallmann, F . W .; Suarez, Pablo Ulises; van Tuyl, November 2017 Andrew H .; van Vilet, Carolyne M . Volume 64, Issue 10 • pages 1129–1248 Solomon Marcus (1925–2016) (Alexandra Bellow) ...... 1216

FEATURES December 2017 AMS Fall Western Sectional Sampler (Paul Balmer, Pavel Etingof, Volume 64, Issue 10 • pages 1249–1376 Monica Vazirani) ...... 1142 Antibiotics Time Machines Are Hard to Build (Ngoc M . Tran, Jed FEATURES Yang, Communicated by Cesar E . Silva) ...... 1136 Ad Honorem Charles Fefferman (Communicated by Stephen COMMUNICATIONS Kennedy, Steven J . Miller) ...... 1254 (An) Old Problem Resurfaces Nonlocally: Gamow's Liquid Drops (The) Concept of Varifold (Ulrich Menne) ...... 1148. Inspire Today's Research and Applications (Rustum Choksi, (A) Glimpse into Discrete Differential Geometry (Keenan Crane, Cyrill B . Muratov, Ihsan Topaloglu) ...... 1275 Max Wardetzky, Communicated by Joel Hass) ...... 1153 Marcel Berger Remembered (Claude LeBrun, Editor and Immigration, Freedom, and the History of the Institute for Translator) ...... 1285 Advanced Study (Fadi Bardawil, Thomas Dodman, Ian Jauslin, Pascal Marichalar, Klaus Oschema, Peter Redfield, COMMUNICATIONS Communicated by Harriet Pollatsek) ...... 1160. Autobiography of Wentsun Wu (1919–2017) ...... 1319. GRADUATE STUDENT SECTION Dorothea Rockburne and Max Dehn at Black Mountain College Blog ...... 1174 (David Peifer) ...... 1313 Interview with Lillian Pierce (Laure Flapan, Communicated by Reducing Bias in Faculty Searches (Elizabeth A . Burroughs) . 1304 Alexander Diaz-Lopez) ...... 1170. GRADUATE STUDENT SECTION WHAT IS…a Varifold? (The Editors) ...... 1173 Blog ...... 1302 OPINION Interview with Ciprian Manolescu (Conducted by Alexander Diaz- Accessible Mathematics for People with Hearing Loss at Colleges Lopez) ...... 1297 and Universities (J . Tilak Ratnanather, Communicated by Steven J . WHAT IS…a Scheme? (Tom Gannon, Communicated by Cesar E . Mills ...... 1180. Silva) ...... 1300 Writing a Teaching Letter (Chris Woodward, Communicated by OPINION Harriet Pollatsek) ...... 1178 Four Goals for Instructors Using Inquiry-Based Learning (Chris BOOK REVIEW Rasmussen, Karen Marrongelle, Oh Nam Kwon, Angie Hodge, (The) Slow Professor (David Manderscheid) ...... 1191 Communicated by Benjamin Braun) ...... 1308 BOOKSHELF ...... 1194 BOOK REVIEW (The) Real and the Complex (Reviewed by Judith V . Grabiner) . 1322 BOOKSHELF ...... 1327

1370 Notices of the AMS Volume 64, Number 11 2017 Index

COMMENTARY LETTERS TO THE EDITOR Article about “Wide Influence” of L . E . Dickson (Stanley Liu) . .1329 Peripatetic Accent Marks (János Kollár) ...... 1329 Updating Math Genealogy ( Della Dumbaugh) ...... 1329 THE BACK PAGE ...... 1330 DEATH NOTICES/MEMORIAMS Marcel Berger Remembered (Claude LeBrun, Editor and Translator) ...... 1285

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December 2017 Notices of the AMS 1371 2017 Index Societal Record

AWARDS AND PRIZES Levi L . Conant Prize, 2017 ...... 324. Bailey, David H . 2016 AAAS Mentor Award ...... 273 Borwein, Jonathan Radunskaya, Ami Mattingly . Andrew 2017 AAAS Fellows Elected ...... 601. Aizenman, Michael Wightwick, Glenn Bhargava, Manjul Ruth Lyttle Satter Prize, 2017 ...... 316 Hacon, Christopher D . DeMarco, Laura Kohn, Robert V . Mirzakhani, Maryam GRANTS, FELLOWSHIPS, AND OPPORTUNITIES Serganova, Vera American Mathematical Society Centennial Fellowship . . . 1208. 2017 AMS Award for Impact on the Teaching and Learning of AMS-AAAS Mass Media Summer Fellowships ...... 1209. Mathematics ...... 474 AMS Congressional Fellowship ...... 1208. Umland, Kristin AMS Epsilon Fund ...... 1209. 2017 AMS Exemplary Program Award ...... 472. AMS-Simons Travel Grants Program ...... 60. The Department of Mathematics at University of Illinois AMS Travel Grants for ICM 2018 ...... 1038 2017 Mathematics Programs That Make a Difference . . . . . 476. Bertrand Russell Prize of the AMS ...... 627 Math Alliance AMS Menger Awards at the 2017 ISEF ...... 927. Math in Scholarships Awarded ...... 1033 First Place (Griffin Macris) Project NExT 2017–2018 ...... 277. Second Place (Savelii Novikov, Matthew Hase-Liu) Ulf Grenander Prize in Stochastic Theory and Modeling ...... 627 Third Place (Egor Morozov, Dmitrii Mikhailovskii, Arjun Ramani, Yuan-Jung Juang) MEETINGS & CONFERENCES Honorable Mention (Dona-Maria Ivanova, Shilun Li, Mas Land, General Information Regarding Meetings & Conferences of the Alec Leng, Gonen Zimmerman) AMS ...... 75 AMS Waldemar J . Trjitzinsky Awards ...... 44. INVITED ADDRESS SPEAKERS BY MEETING Adonteng, Renee ANN ARBOR, MICHIGAN: Fuchs, Elena; Putman, Andrew; Smart, Coleman, Andrew D . Charles ...... 669 Coelho, Terence ATLANTA, GEORGIA: Colding, Tobias Holck; Kenig, Carlos E .; Dupont, Madison C . Preskill, John; Simon, Barry; Staffilani, Gigliola; Taylor, Richard; Emblem, Seth Erik Wienhard, Anna ...... 77 Garcia, Edwin BLOOMINGTON, INDIANA: Demeter, Ciprian; Koch, Sarah C ;. Leach, Breana Kechelle Schwartz, Richard Evan ...... 85 Bôcher Memorial Prize, 2017 ...... 317 BOSTON, MASSACHUSETTS: Ding, Jian; Frenkel, Edward; Tosatti, Vasy, András Valentino; Viazovska, Maryna ...... 667. Centennial Fellowship Award, 2017–2018 ...... 595 BUFFALO, NEW YORK: Kim, Inwon C .; Menon, Govind; Sagan Bruce ...... 91. Takeda, Shuichiro CHARLESTON, SOUTH CAROLINA: Achar, Pramod N ;. Bray, Hubert; Frank and Brennie Morgan Prize for Outstanding Research in Chertock, Alina ...... 80. Mathematics by an Undergraduate Student, 2017 . . . . . 330 Yang, David H . COLUMBUS, OHIO: Brown, Aaron; Dymarz, Tullia; Huh, June . . 666 Frank Nelson Cole Prize in Number Theory, 2017 ...... 319 DENTON, TEXAS: Çiperiani, Mirela; Gillman, Adrianna; Pilgrim, Kevin ...... 90. Darmon, Henri FAYETTEVILLE, ARKANSAS: Dafermos, Mihalis; Hauenstein, Joseph L . Doob Prize, 2017 ...... 327. Jonathan; Mann, Kathryn ...... 668 Friedlander, John HONOLULU, HAWAII: Mazur, Barry; Naber, Aaron; Needell, Iwaniec,Henryk Deanna; Stange, Katherine; Suk, Andrew ...... 1111 Leonard Eisenbud Prize for Mathematics and Physics, 2017 . . .321 NASHVILLE, TENNESSEE: Bertozzi, Andrea; Landsberg, J . M ;. Erd˝os, László Morse, Jennifer; Wickelgren, Kirsten ...... 667. Yau, Horng-Tzer NEWARK, DELAWARE: Greengard, Leslie; Grigsby, Elisenda; Maulik, Leroy P . Steele Prizes, 2017 ...... 311 Davesh ...... 668 Arthur, James NEW YORK, NEW YORK: Kahn, Jeremy; Marques, Fernando Codá; McDuff, Dusa Maynard, James; Ramanan, Kavita ...... 87 Salamon, Dietmar ORLANDO, FLORIDA: Heitsch, Christine; Kujawa, Jonathan; Sogge, Simon, Leon Christopher D ...... 91

1372 Notices of the AMS Volume 64, Number 11 2017 Index

PORTLAND, OREGON: Kovács, Sándor; Mantovan, Elena; AMS–AAAS Mass Media Summer Fellowships ...... 1331 Shlyakhtenko, Dimitri ...... 428 AMS Department Chairs Workshop Held ...... 514, 1335 PULLMAN, WASHINGTON: Hitrik, Michael; Raich, Andrew S .; AMS E-mail Support for Frequently Asked Questions ...... 160 Rogalski, Daniel ...... 86. AMS Employment Center ...... 1041 RIVERSIDE, CALIFORNIA: Balmer, Paul; Etingof, Pavel; Vazirani, Monica ...... 92 AMS Graduate Student Chapter Spotlight ...... 568 SAN DIEGO, CALIFORNIA : Ardila, Federico; Bryant, Robert; AMS Holds Congressional Briefing on Capitol Hill ...... 514 Charney, Ruth; Dwork, Cynthia; Randall, Dana; Titi, Edriss; Wigderson, Avi ...... 798 AMS-MSRI Co-sponsored Congressional Briefing ...... 918 SAN FRANCISCO, CALIFORNIA : Iyengar, Srikanth B .; Witherspoon, AMS Short Course ...... 1016 Sarah; Yakubu, Abdul-Aziz ...... 669 AMS–Simons Travel Grants Program ...... 60. SHANGHAI, PEOPLE'S REPUBLIC OF CHINA: Dai, Yu-Hong; Ribet, AMS-Sponsored CNSF Exhibition ...... 918 Kenneth A .; Schoen, Richard M ;. Wu, Sijue; Xu, Chenyang; You, Jiangong ...... AMS Staff Support Local K–12 Education ...... 623 ...... 668 Call for Nominations AMS Exemplary Program Award . . . . 447 2017–2018 MEETING ANNOUNCEMENTS AND PROGRAMS Call for Nominations AMS Award for Mathematics Programs That Make a Difference ...... 501 Atlanta, Georgia (2017 JMM) ...... 77 Call for Nominations Award for Impact on the Teaching and Bloomington, Indiana (2017) ...... 176 Learning of Mathematics ...... 455 Buffalo, New York (2017) ...... 652. Call for Nominations AWM–AMS Noether Lecture ...... 708 Charleston, South Carolina (2017) ...... 80. Departments Coordinate Job Offer Deadlines ...... 1331 Denton, Texas (2017) ...... 646 Epsilon Awards Announced ...... 622. Montréal, Quebec, Canada (2017) ...... 88. Erd˝os Memorial Lecture ...... 62, 623 New York, New York (2017) ...... 291 Fan China Exchange Program Grants Awarded ...... 622. Orlando, Florida (2017) ...... 657. Pullman, Washington (2017) ...... 181 Free Grant-Writing Workshop Offered ...... 1335 Riverside, California (2017) ...... 951 From the AMS Public Awareness Office . . . . 399, 765, 918, 1033 2017–2021 MEETINGS JMM 2017 Photo Key and Montage ...... 359, 360 Ann Arbor, Michigan (2018) ...... 189 JMM 2018 Meeting Registration Forms ...... 1127, 1128, 1247 Auburn, Alabama (2019) ...... 534 JMM 2017 Program at a Glance ...... 1113 Baltimore, Maryland (2019) ...... 94. (The) Line Newsletter Connects Supporters and Beneficiaries . 919 Binghamton, New York (2019) ...... 1246 Math in Moscow Scholarship Program ...... 395 Boston, Massachusetts (2018) ...... 93. Buffalo, New York (2017) ...... 91 Mathematical Moments: Designing Better Bicycles ...... 887 Columbus, Ohio (2018) ...... 92 Mathematical Moments: Maintaining a Balance ...... 137 Denton, Texas (2017) ...... 90. Mathematics Research Communities 2018 ...... 1039. Denver, Colorado (2020) ...... 94 Member Spotlight (Darryl H . Yong) ...... 725. Fayetteville, Arkansas (2018) ...... 189 New AMS Blog from the Director of the AMS Washington Office Honolulu, Hawaii (2019) ...... 94...... 623 Nashville, Tennessee (2018) ...... 93 “News from the Road” ...... 399 Newark, Delaware (2018) ...... 93 Project NExT 2017 Fellows Chosen ...... 1335 Orlando, Florida (2017) ...... 91 Organizing the 2019 AMS Short Course ...... 1039. Portland, Oregon (2017) ...... 93. Portland, Oregon (2018) ...... 188 Report of the Executive Director: State of the AMS, 2016 . . . 914 Quy Nhon City, Vietnam (2019) ...... 1111 Report of the AMS Treasurer (2016) ...... 10. 26 Riverside, California (2017) ...... 92 Start an AMS Graduate Student Chapter at Your Institution . . 465 San Diego, California (2018) ...... 188 Women Doing Mathematics ...... 250 San Francisco, California (2018) ...... 94. NEWS Shanghai, People’s Republic of China (2018) ...... 93 Inside the AMS . . . . .62, 160, 399, 514, 622, 918, 1033, 1217, 1336 Washington, DC (2021) ...... 94 FYI ...... 160, 1338 THE AMS Mathematics Opportunities . . .60, 156, 395, 627, 920, 1208, 1331 American Mathematical Society Centennial Fellowship . . . . 920 Mathematics People . . . . .52, 147, 272, 392, 595, 924, 1211, 1334 AMS Congressional Fellow Announced ...... 764 FROM THE SECRETARY'S OFFICE AMS–AAAS Mass Media Fellow Chosen ...... 764 2016 AMS Election Results ...... 142

December 2017 Notices of the AMS 1373 2017 Index

2017 AMS Election: Nomination by Petition ...... 144 Reciprocity Agreements ...... 1198

2017 American Mathematical Society Elections Report of the Executive Director: State of the AMS, 2016 . . . .914 Table of Contents ...... 837 Report of the AMS Treasurer, 2018 ...... 1026 List of Candidates ...... 838 Statistics on Women Mathematicians ...... 994 Election Information ...... 838 Voting Information for 2017 AMS Election (Carla D . Savage) . . 711 Proposed Amendments to the Bylaws ...... 840 NEW PUBLICATIONS OFFERED BY THE AMS . . . 63, 167, 278, Nominations for President ...... 842 409, 517, 631, 786, 937, 1048, 1219, 1340 Biographies of Candidates 2017 ...... 845 Current Bylaws ...... 860 SURVEYS AND REPORTS Call for Suggestions, 2018 ...... 865,. 1141 2016–2017 Faculty Salaries Report (Amanda L . Golbeck, Thomas H . Nominations by Petition ...... 866 Barr, and Colleen A . Rose) ...... 906 Doctoral Degrees Conferred ...... 727 2017 AMS Award for Impact on the Teaching and Learning of Mathematics ...... 474 Report on 2015–2016 Academic Recruitment, Hiring, and Attrition 2017 AMS–MAA–SIAM Frank and Brennie Morgan Prize for (Amanda L . Golbeck, Thomas H . Barr, and Colleen A . Rose).. 584 . Outstanding Research in Mathematics by an Undergraduate Student ...... 330 2017 Award for an Exemplary Program or Achievement in a Mathematics Department ...... 472. 2017 Award for Mathematics Programs That Make a Difference ...... 476. 2017 Bôcher Memorial Prize ...... 317 2017 Class of Fellows of the AMS ...... 268 2017 Frank Nelson Cole Prize in Number Theory ...... 319 2017 Joint Policy Board for Mathematics Communications Awards ...... 479 2017 Joseph L . Doob Prize ...... 327 2017 Leonard Eisenbud Prize for Mathematics and Physics . . . 321 2017 Leroy P . Steele Prizes ...... 311 2017 Levi L . Conant Prize ...... 324 2017 Ruth Lyttle Satter Prize ...... 316 2018 Class of Fellows of the AMS: Call for Nominations . . . . 238. 2018 Class of Fellows of the AMS Selected ...... 1217 AMS 2016 Contributors List ...... 488 AMS Governance, 2017 ...... 625 Backlog of Mathematics Research Journals ...... 1184 Biennial Overview of AMS Honors (Carla D . Savage) . . . . . 1200 Call for Nominations for Albert Leon Whiteman Memorial Prize, Award for Distinguished Public Service, Chevalley Prize in Lie Theory, Frank Nelson Cole Prize in Algebra, George David Birkhoff Prize in Applied Mathematics, Levi L . Conant Prize ...... 390, 486 Call for Nominations for D . R . Fulkerson Prize ...... 12. 74 Call for Nominations for Leroy P . Steele Prizes ...... 141, 195 New AMS Member Benefit ...... 1217 NEW Opportunity: Ulf Grenander Prize . . . 398, May Inside Cover NEW Opportunity: Bertrand Russell Prize of the AMS . . 400, 495

1374 Notices of the AMS Volume 64, Number 11 2018 Joint Mathematics Meetings Advance Registration/Housing Form

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I DO NOT want my badge and program (if printed program is requested) to be mailed to me on 12/8/17. Materials will be mailed to the address listed above unless you check this box. Materials will not be mailed for registrations completed after November 22, or to individual commercial or artist exhibitors. Exhibiting companies may opt to have booth staff badges mailed to a company contact. Registration Fees Payment Membership please P all that apply. First row is eligible to register as a member. Registration & Event Total (total from column on left) $ For undergraduate students, membership in PME and KME also applies. o AMS & MAA o AMS but not MAA o MAA but not AMS o ASL o CMS o SIAM Hotel Deposit (only if paying by check) $ Undergraduate Students Only: o PME o KME If you send a hotel deposit check, the deadline for this form is December 5. Other Societies: o AWM o NAM o YMN o AMATYC Total Amount To Be Paid $ Joint Meetings by Dec 20 at mtg Subtotal o Member AMS, MAA, ASL, CMS, or SIAM US$ 329 US$ 433 Method of Payment o Nonmember US$ 522 US$ 666 o Check. Make checks payable to the AMS. For all check payments, o Graduate Student Member (AMS, MAA please keep a copy of this form for your records. ASL, CMS, or SIAM) US$ 74 US$ 86 o Credit Card. All major credit cards accepted. For your security, we o Graduate Student (Nonmember) US$ 118 US$ 130 do not accept credit card numbers by email, fax, or postal mail. If the MMSB o Undergraduate Student (Member AMS, receives your registration form by any of these methods, it will contact you at ASL, CMS, MAA, PME, KME, or SIAM) US$ 74 US$ 86 the phone number provided on this form. o Undergraduate Student (Nonmember) US$ 118 US$ 130 o High School Student US$ 7 US$ 14 Signature: o Unemployed US$ 74 US$ 86 o Temporarily Employed US$ 268 US$ 307 o Purchase Order # (please enclose copy) o Developing Countries Special Rate US$ 74 US$ 86 o Emeritus Member of AMS or MAA US$ 74 US$ 86 Other Information o High School Teacher US$ 74 US$ 86 o Librarian US$ 74 US$ 86 Mathematical Reviews field of interest # o Press US$ 0 US$ 0 o I am willing to serve as a judge for the MAA Undergraduate Student o Exhibitor (Commercial) US$ 0 US$ 0 Poster Session o Artist Exhibitor (work in JMM Art Exhibit) US$ 0 US$ 0 o For planning purposes for the MAA Two-year College Reception, please o Nonmathematician Guest check if you are a faculty member at a two-year college. of registered mathematician US$ 21 US$ 21 o I am a mathematics department chair. $ o Please do not include my name and postal address on any promotional AMS Short Course: Random Growth Models (1/8-1/9) mailing lists. (The JMM does not share email addresses.) o Member of AMS US$ 114 US$ 148 o Please do not include my name on any list of JMM participants other than o Nonmember US$ 175 US$ 205 the scientific program if I am, in fact, making a presentation that is part of o Student, Unemployed, Emeritus US$ 62 US$ 83 the meeting. $ o Please P this box if you have a disability requiring special services. MAA Minicourses (see listing in text) Registration for the Joint Meetings is not required for the short course but it is required for I would like to attend: o One Minicourse o Two Minicourses the minicourses and the Employment Center. To register for the Employment Center, go Please enroll me in MAA Minicourse(s) #______and #______to http://www.ams.org/profession/employment-services/employment-center. For questions, Price: US$ 100 for each minicourse. email: [email protected]. (For more than 2 minicourses, call or email the MMSB.) $ Registration Deadlines Graduate School Fair Table To be eligible for the complimentary hotel room lottery: Oct. 31, 2017 o Graduate Program Table US$125 US$125 In time to receive badges/programs in the mail: Nov. 22, 2017 (includes table, posterboard & electricity) Dept. or Program to be represented (write below or email) Hotel reservations with check deposit: Dec. 5, 2017 $ Hotel reservations, changes/cancellations Receptions & Banquets through the JMM website: Dec. 6, 2017 o Graduate Student/First-Time Attendee Reception (1/10) (no charge) Advance registration for the Joint Meetings, short o NAM Banquet (1/12) course, minicourses, and dinner tickets: Dec. 20, 2017 #____Chicken #____Fish #____Vegan US$ 75 Cancel in time to receive 50% refund on advance #____Kosher (Additional fees apply for Kosher Meals. ) US$ 125 registration, banquets, minicourses, and short course Jan. 4, 2018* Total for NAM Banquet $ *no refunds issued after this date. o AMS Dinner (1/13) Regular Price #____US$ 75 Student Price #____US$ 30 Mailing Address/Contact: (For special dietary requests, please email [email protected]) Mathematics Meetings Service Bureau (MMSB) Total for AMS Dinner $ P. O. Box 6887 Providence, RI 02940-6887 Fax: 401-455-4004; Email: [email protected] Total for Registrations and Events $ Telephone: 401-455-4144 or 1-800-321-4267 x4144 or x4137 2018 Joint Mathematics Meetings Hotel Reservations – San Diego, CA

Please see the hotel information in the announcement or on the web for detailed information on each hotel. To ensure accurate assignments, please rank hotels in order of preference by writing 1, 2, 3, etc. in the column on the left and by circling the requested bed confirguration. If your requested hotel and room type is no longer available, you will be assigned a room at the next available comparable rate. Please call the MMSB for details on suite configurations, sizes, availability, etc. All reservations, including suite reservations, must be made through the MMSB to receive the JMM rates. Reservations made directly with the hotels before December 15, 2017 may be changed to a higher rate. All rates are subject to applicable local and state taxes in effect at the time of check-in; currently 10.5% state tax, the San Diego Tourism Marketing District assessment 2% tax, and the CA Tourism fee of US$0.77 per night. Guarantee requirements: First night deposit by check (add to payment on reverse of form) or a credit card guarantee. Please note that reservations with check deposits must be received by the MMSB by December 5, 2017. People interested in suites should contact the MMSB directly at [email protected] or by calling 800-321-4267, ext. 4137; (401-455-4137). o Deposit enclosed (see front of form) o Hold with my credit card. For your security, we do not accept credit card numbers by email, postal mail or fax. If the MMSB receives your registration form by any of these methods, it will contact you at the phone number provided on the reverse of this form.

Date and Time of Arrival Date and Time of Departure Number of adult guests in room Number of children

Name of Other Adult Room Occupant (s) Arrival: Departure:

Housing Requests:.(example: rollaway cot, crib, nonsmoking room, low floor) o I have disabilities as defined by the ADA that require a sleeping room that is accessible to the physically challenged. My needs are: o I am a member of a hotel frequent-travel club and would like to receive appropriate credit. The hotel chain and card number are: o I am not reserving a room. I am sharing with , who is making the reservation.

Order Hotel Single Double Double Triple Quad Rollaway/Cot Fee of choice 1 bed-2 people 2 beds- 2 people 3 adults-2 beds 4 adults-2 beds (add to special requests if reserving online) Marriott Marquis San Diego Marina (hdqrs) Bay View US$ 222 US$ 222 US$ 222 US$ 242 UD$ 262 Rollaways are available (at no charge) in king-bedded rooms only City View US$ 207 US$ 207 US$ 207 US$ 227 UD$ 247 City View Student Rate US$ 147 US$ 147 US$ 147 US$ 167 UD$ 187 Embassy Suites Hotel San Diego Bay US$ 180 US$ 180 US$ 180 US$ 200 US$ 220 Rollaways are not available. Student Rate US$ 160 US$ 160 US$ 160 US$ 180 US$ 200 Manchester Grand Hyatt San Diego US$ 179 US$ 189 US$ 189 US$ 199 US$ 209 Rollaways are available (at no charge) in king-bedded rooms only. Student Rate US$ 147 US$ 147 US$ 147 US$ 157 US$ 167 Omni Hotel San Diego US$ 179 US$ 189 US$ 189 US$ 199 US$ 209 Rollaways are available (at no charge) in king-bedded rooms only Student Rate US$ 147 US$ 147 US$ 147 US$ 157 US$ 167 Hilton Gaslamp San Diego US$ 176 US$ 176 US$ 176 US$ 176 US$ 176 Maximum 4 people per room. Rollaways are available for US$ 20 per day in king-bedded rooms only. Student Rate US$ 158 US$ 158 US$ 158 US$ 158 US$ 158 Hard Rock Hotel San Diego US$ 170 US$ 170 US$ 170 US$ 190 US$ 210 Rollaways are available for US$ 30 per day in king- bedded rooms only. Student Rate US$ 159 US$ 159 US$ 159 US$ 179 US$ 199 Best Western Plus Bayside Inn US$ 165 US$ 165 US$ 165 US$ 175 US$ 175 Rollaways are available on request for US$ 10 per day. Student Rate US$ 155 US$ 155 US$ 155 US$ 165 US$ 165 Solamar Hotel San Diego US$ 165 US$ 165 US$ 165 US$ 185 US$ 205 Rollaways are available for US$ 40 per day in king- bedded rooms only. Student Rate US$ 149 US$ 149 US$ 149 US$ 169 US$ 189 Palomar Hotel San Diego US$ 160 US$ 160 US$ 160 US$ 185 US$ 210 Rollaways are not available. Student Rate US$ 149 US$ 149 US$ 149 US$ 174 US$ 199 Horton Grand Hotel US$ 159 US$ 159 US$ 159 US$ 179 US$ 199 Rollaways are not available. Student Rate US$ 139 US$ 139 US$ 139 US$ 159 US$ 179 Porto Vista Hotel US$ 125 US$ 125 US$ 125 US$ 145 US$ 165 Rollaways are not available. IN THE NEXT ISSUE OF NOTICES

JANUARY 2018…

“We all have a responsibility to inspire a new generation in STEM and nurture the dreams of future mathematical leaders.” —Talithia Williams

At the 2018 Joint Mathematics Meetings, Talithia Williams will be delivering the AMS-MAA-SIAM Hrabowski-Gates-Tapia-McBay Lecture “Mathematics for the Masses,” where she discusses the new six-part PBS series NOVA Wonders, for which she is one of the hosts. Read Talithia’s introduction to her lecture in the January Notices JMM 2018 Lecture Sampler. Also included in the sampler are lecture introductions from: Federico Ardila; Gunnar Carlsson; Ruth Charney; André Neves; Ronald E. Mickens; Jill Pipher; Dana Randall; and Erica Walker.

Are you looking for holiday gift ideas at a great price?

Visit bookstore.ams.org to browse our Holiday Sale. Select titles are discounted until 12/17/17. American Mathematical Society Distribution Center 35 Monticello Place, Pawtucket, RI 02861 USA

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