Natalie Zemon Davis, et al., “Remembrances of Chandler Davis (1926–2022),” Notices of the American Mathematical Society, vol. 70, no. 7 (August 2023): 1107-1119, esp. 1113

Remembrances of Chandler Davis (1926–2022) Sheldon Axler, John J. Benedetto, Rajendra Bhatia, Man-Duen Choi, Aaron Davis, Natalie Zemon Davis, Simone Weil Davis, John Friedlander, Mary Gray, Stephen Halperin, Peter Rosenthal, Marjorie Senechal, and Hannah Davis Taïeb Figure 1. Chandler Davis, 2018. Peter Rosenthal In 1960, I heard a radio interview with an extraordinary man. He was a mathematician and was also a science fiction writer. He was about to be imprisoned because of his Peter Rosenthal is a professor emeritus of mathematics at the University of Toronto. His email address is rosent@math.utoronto.ca. For permission to reprint this article, please contact: reprint-permission@ams.org. refusal to testify in front of the House Un-American Activities Committee (HUAC). His interview was articulate, eloquent, and reflected a deep commitment to his beliefs. Later, when I came to know him well, I understood what an inspiring and principled man he was. The man was Horace Chandler Davis (widely known as “Chandler Davis” or “Chan Davis”). On September 24, 2022, he passed away in Toronto at the age of 96 from a probable stroke. Chandler was a wonderful husband, father, and grandfather, an excellent mathematician, an extremely active political activist, an author of very interesting science fiction stories, a staunch feminist, and a fine poet and composer. He never seemed defeated by or bitter about the obstacles he encountered. He worked tirelessly towards a more egalitarian world, participating in many progressive activities throughout his long life. Chandler was born on August 12, 1926 in Ithaca, New York, the eldest of five children of Marian R. Davis and Horace Bancroft Davis. His parents were economists whose political views were very left-wing. Like Chan, his father was fired from his position at a university because he refused to answer questions asked by HUAC. Chandler received his PhD in mathematics from Harvard University in 1950 and became an instructor in the Department of Mathematics at the University of Michigan. DOI: https://doi.org/10.1090/noti2740 AUGUST 2023 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1107 In 1953, Chandler was subpoenaed to be a witness before the House Un-American Activities Committee, which investigated allegations of communist activity in the United States. Chandler refused to answer the committee’s questions. Unlike most uncooperative witnesses, he invoked the First Amendment of the United States Constitution, which guarantees free speech, rather than using the Fifth Amendment’s protection against self-incrimination. Chandler wanted to establish a precedent that HUAC had no right to ask witnesses questions about their political beliefs. He knew that he risked being cited for contempt of Congress and sent to jail, but he wanted to raise awareness of the dangers of HUAC. Chandler was then fired by the University of Michigan. On December 3, 1959, the Supreme Court refused to hear his case. Chandler surrendered to serve six months in federal prison. Chandler continued his research in mathematics before, during, and after his incarceration. He retained his sense of humour throughout: A footnote to a mathematics paper [Dav63a] that he wrote while incarcerated reads: Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author’s and are not necessarily those of the Bureau of Prisons. During the several years between his dismissal from Michigan and his imprisonment, Chandler applied for many different positions. It became apparent that he was blacklisted. The blacklist continued even after Chandler got out of prison in 1960. In 1962, with support from the distinguished Canadian mathematician H.S.M. Coxeter, Chandler accepted a position as professor of mathematics at the University of Toronto. Chan flourished at U of T. He was an excellent teacher, supervised fifteen PhD theses and continued to make significant research contributions to mathematics, especially to linear algebra and operator theory. Chan’s teaching inspired many students to become mathematicians. James Arthur, a University Professor and Mossman Chair at the University of Toronto who served a term as president of the American Mathematical Society, writes: Chandler Davis was my colleague for over forty years. I admired him greatly. His course in real and complex analysis, which I took as a third-year undergraduate at Toronto, was a transformative experience for me, and, I would say, for every other student in the course. Chandler Davis was a left-wing radical who participated in a huge number of progressive causes, both on 1108 campus and off, throughout his long life. Chandler opposed the American–Vietnamese war and was chairman of the Toronto Anti-Draft Program. He was active in Science for Peace and often participated in the Toronto Vigil against the Occupation of the Territories. He regularly attended the Davis, Markert, Nickerson Lecture in Academic and Intellectual Freedom, established in the 1990s by the University of Michigan Faculty Senate in answer to the University’s treatment of faculty, including Chandler, who had been attacked by HUAC. A few weeks before his death, Chandler co-organized and spoke from his hospital bed at an online event in support of imprisoned dissident Russian mathematician Azat Miftakhov. Chan began his talk as follows: It is a pleasure to welcome you to this panel in support of our young colleague Azat Miftakhov and other political prisoners; in support, in particular, of Russians courageously speaking out against the war, and, more generally, in support of freedom of conscience and peace. It means a lot to me to be opening this session because I have a special bond to Azat Miftakhov. I was a political prisoner myself, years ago, not in Russia but in the USA. I was not much older than he is now; like him I had a wife standing by me outside; and like him I tried to go ahead doing mathematics while in prison. It was hard, but not as hard as Azat’s imprisonment, and it was only half a year. Chandler often raised political issues within the community of mathematicians. Many mathematicians resented such activities, arguing that it was wrong to “politicize” mathematics. Chandler was a very staunch feminist (see the contribution from Mary Gray later in this article). Chan and his wife, the distinguished historian Natalie Zemon Davis, agreed that their marriage would be based on gender equality. They shared care of their three children, even during periods of their lives when they held professorships at universities on opposite sides of North America, Chan at Toronto and Natalie at Berkeley and then Princeton. When Chan turned 65, he was mandatorily retired and became a professor emeritus. That did not change his life very much. He still maintained his research, taught some courses, and supervised PhD students. He continued to serve as editor-in-chief of the Mathematical Intelligencer. In 2010, Josh Lukin compiled and edited It Walks in Beauty, a compilation of some of Chan’s essays and stories. This book is in the Aqueduct Press series of Heirloom Books, which aims to bring back into print and preserve work that has helped make feminist science fiction what it is today. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 70, NUMBER 7 John J. Benedetto Chandler arrived in Toronto in 1962. I arrived in Toronto in 1962 with an MA from Harvard and having been raised in a working class suburb of Boston. So, at that point we had a little bit of Harvard in common, but not much else. I taught a section of calculus and Chandler was in charge. I became his PhD student the very first day we met—a separate story. It was the wisest, perhaps luckiest, happening of my mathematical life. Chan seemed old to me. I had just turned 23, and he was 36. I had already decided on Laplace transforms, topological vector spaces, and Schwartz’s theory of distributions for a general thesis area. We met every week in the adviser/advisee dance. He taught me all of the things he could, that I could understand. I loved those meetings, and learned so much from him in them; but, dutifully, as any rebellious child would behave, I did nothing about it at the time. This was a real error on my part, since in later years I understood more and more how deep and ingenious and knowledgeable he was. When Chandler told me about Naimark’s theorem and its importance and his creative contributions in this area, and all the wonderful mathematics he knew and did, I should have pursued all of it more actively. In fact, so many of his ideas and contributions play a major role in the theory of frames that I have been working on for 25 years. Frames, going back to Paley and Wiener, and then Duffin and Schaeffer, reemerged in the early 1990s as a vehicle for extending and applying wavelet and timefrequency systems both in terms of generality and genuine applicability. My weekly meetings with Chandler were replete with the linear algebra and operator theory necessary for such generality and applicability. I learned so much from Chandler, without doing much about it when I was a graduate student. However, I knew from the beginning that Chandler was brilliant. What certainly stuck from our meetings were the breadth and overall appreciation and excitement of mathematics—I cannot imagine a better adviser. In any case, he let me run where I wanted to go, and filled in mathematical gaps prodigiously—a protective father, who kept me on schedule. Early on in Toronto, I found out Chandler was a legend in a cause celebré. His mathematics paper [Dav63a] was written while he was incarcerated as a result of his courageous testimony to the House Un-American Activities Committee. His red badge of courage (sic) (and John J. Benedetto is Research and Emeritus Professor of mathematics at the University of Maryland and the founding director of the Norbert Wiener Center. His email address is jjb@umd.edu. AUGUST 2023 humor) in this paper became a badge of pride and honor for me. In [Dav63a], MY adviser thanked the Federal Prison System in the acknowledgments, further noting that his opinions there were not necessarily those of the Bureau of Prisons. My fellow North American graduate students only had advisers who could acknowledge national scientific support organizations. Wow, I was so fortunate! By the way, in [Dav63a] Chan solved the so-called second Ungar conjecture. Knowing, I suppose, that I had pried into his past, he told me that his political activism was a thing of the past. Thank goodness he had second thoughts. He was brave and so principled, and it was his lifelong mantra. Our relation evolved and deepened over the years; and this is a beautiful experience I’ve had with many of my own graduate students. In the process, I learned so many things about the political environment that I still try to understand, and of which I am so bewildered. Most important we’ve had a decades-long correspondence on such matters, all to my benefit. So when Schrecker’s No Ivory Tower [Sch86] appeared in 1986, along with the many reviews, I pounced on it and some of them. It was authoritative and gripping and compelling. And then Chandler’s own The Purge [Dav88] appeared. I knew some of these folks in that article! Raoul Bott and Ed Moise were my instructors at Harvard. Hans Lewy and I became friends through daily lunches and regular dinners while on our sabbaticals at the Scuola Normale Superiore in Pisa. And Lee Lorch and I became friends through Ray Johnson. Lucky me, reminding me of what Chan wrote in The Purge in a different context: “The experience of marginality is good for the soul and better for the intellect.” I never met Nate Coburn, whom Chan also highlighted in the The Purge, but his son Lew and I were office mates at NYU in 1964–1965 and are good friends. Naturally, I met Natalie and was dazzled. What a couple—beyond anything I had ever imagined, and a diamond anniversary love affair. Her gift to Chandler on his 60th birthday was a book of his poems (Having Come This Far, 1986) selected by Natalie and their daughters, Hannah and Simone. Their son, Aaron, had set some of the poems to music. I just reread the poems. They are mostly beautiful and loving, perceptive and lyrical; and they are about Natalie and their family, but also about the poolplaying freedom rider and there is the eccentric seafood song and everything in between. Besides an amazing career, Natalie entertained graduate students in Toronto, e.g., me, and she and Chandler stayed with Cathy and me in Maryland. We had been in correspondence, and when Chandler died, she wrote that several days earlier she and Chandler found out that they would be having another NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1109 great-grandchild. She added that “life continues, with sorrow and with hope.” Chandler introduced me to Laurent Schwartz when Schwartz visited Toronto. The three of us had a memorable (for me) lunch together. Close to graduation time, Chan gave some fatherly insights to me. He noted that I was not getting any younger (I was 24 when I received my PhD in 1964); and therefore I should work very hard. Lest this body blow was not sufficient, he also noted that an outsider might construe that anything in my thesis of worth was due to my adviser; and therefore I should work very hard. Chan was a very subtle fellow! His telephone call to NYU got me a tenure-track position. Chandler has continued to be my hero through all the years, whether it was because of his poetry, his principles, or his mathematics—I never did read his science fiction. At a mathematical fest at the University of Maryland in 1999, he was virtuosic and humble and original and thoughtful as always. In 2019, he was scheduled to speak at another mathematical fest at Maryland. A month before the event, he wrote: “Whom am I kidding, John. I just can’t travel;” at the same time he wrote a long letter that I treasure. And then I was going to visit Toronto in May 2022, and I, too, had to cancel because I couldn’t travel. Alas. Bottom line and my last line: Chandler was extraordinary, and I am truly proud to be his student! Then the Davis-Kahan theorem is the inequality ‖𝐸𝐹‖ ≤ 1 ‖𝐴 − 𝐵‖. 𝛿 (1) This inequality captures several essential features of the problem. When 𝐴 = 𝐵, the projections 𝐸 and 𝐹 are mutually orthogonal (because the eigenvectors of 𝐴 corresponding to distinct eigenvalues are mutually orthogonal). So ‖𝐸𝐹‖ = 0. If 𝐴 is close to 𝐵, one might expect ‖𝐸𝐹‖ to be close to 0. The inequality (1) is a realisation of this. The dependence on 𝛿 is dictated by several examples. Elegant in formulation and powerful in applications, the DavisKahan theorem is one of the best-known results in numerical linear algebra. Among other things, Davis and Kahan recognized the connections between this problem and another involving the Sylvester equation 𝐴𝑋 − 𝑋𝐵 = 𝑌 , of great importance in several areas [BR97]. When the spectra of 𝐴 and 𝐵 are disjoint, this equation always has a unique solution 𝑋 for every 𝑌 . The problem is to find good bounds for the solution 𝑋. If 𝐴 and 𝐵 are hermitian, and the spectrum of 𝐴 Rajendra Bhatia Although the behaviour of eigenvalues of a hermitian matrix under perturbation is well understood, there has been almost nothing done on the behaviour of the eigenvectors. It is well known that they vary analytically under analytic perturbations, but for some purposes one would prefer sharp bounds on the distance between the eigenvectors of a matrix and those of a matrix approximating it. —From the opening paragraph of Chandler Davis’s paper [Dav63b]. With admirable clarity Chandler Davis set himself the goal of finding such sharp bounds. His efforts culminated in the famous “sin 𝜃 theorem” of Davis and Kahan [DK70]. Let 𝐴 and 𝐵 be two hermitian matrices. Let 𝐸 be the eigenprojection of 𝐴 corresponding to its eigenvalues lying inside an interval [𝛼, 𝛽], and 𝐹 the eigenprojection of 𝐵 corresponding to its eigenvalues lying outside (𝛼 − 𝛿, 𝛽 + 𝛿). Rajendra Bhatia is a professor in and the head of the mathematics department at Ashoka University. His email address is rajendra.bhatia@ashoka.edu .in. 1110 Figure 2. Rajendra Bhatia and Chandler Davis at the Grand Canyon, 1989. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 70, NUMBER 7 lies inside [𝛼, 𝛽] and that of 𝐵 outside (𝛼 − 𝛿, 𝛽 + 𝛿), then ‖𝑋‖ ≤ 1 ‖𝐴𝑋 − 𝑋𝐵‖, 𝛿 ‖𝑋‖ ≤ 𝑐1 ‖𝐴𝑋 − 𝑋𝐵‖, 𝛿 (2) and the inequality (1) can be derived from this [DK70]. The spectra of 𝐴 and 𝐵 need to be separated in a rather special way for the inequalities (1) and (2) to hold. If 𝐾1 and 𝐾2 are two arbitrary subsets of the real line with dist(𝐾1 , 𝐾2 ) = 𝛿 > 0 and 𝐸, 𝐹 the eigenprojections of 𝐴, 𝐵 corresponding to them, then these inequalities break down. This was noted by Davis and Kahan, and the first open question posed by them was what best could be said in this case. In [BDM83] Chandler returned to this question with new collaborators to provide a decisive answer. They showed that there exists a universal constant 𝑐1 (independent of the dimension of 𝐴 and 𝐵) such that instead of (2) we have and a similar inequality holds in place of (1). Further, these authors showed that 𝑐1 < 2. This was achieved by obtaining a new form of solution of the Sylvester equation expressed as a Fourier integral and then expressing 𝑐1 as the solution of a minimal extrapolation problem for the Fourier transform. Unknown to the authors, this problem for the Fourier transform had been considered earlier in a totally different context (number theory), where it had been shown by Sz.-Nagy [SN53] that 𝑐1 = 𝜋/2. The authors of [BDM83] also considered an analogue of these problems when 𝐴 and 𝐵 are normal matrices. Now 𝐾1 and 𝐾2 are subsets of the complex plane with dist(𝐾1 , 𝐾2 ) = 𝛿 > 0. In this case they showed that there exists a constant 𝑐2 such that 𝑐 ‖𝑋‖ ≤ 2 ‖𝐴𝑋 − 𝑋𝐵‖, 𝛿 where 𝑐2 is the solution of a minimal extrapolation problem for the Fourier transform in the plane. This turns out to be a harder problem than that of determining 𝑐1 . It was shown in [BDK89] that 𝑐2 < 2.91. (Later Hormander and Bernhardsson [HB93], in a completely different context showed that 2.903887282 < 𝑐2 < 2.9038872828.) This estimate for eigenvectors led to major progress on a longstanding problem about eigenvalues. In 1912, H. Weyl had shown that the eigenvalues of hermitian matrices 𝐴 and 𝐵 can be enumerated as 𝛼1 , … , 𝛼𝑛 and 𝛽1 , … , 𝛽𝑛 in such a way that max |𝛼𝑗 − 𝛽𝑗 | ≤ ‖𝐴 − 𝐵‖. (3) For several years many mathematicians tried to prove that the same result would be true for normal matrices 𝐴 and 𝐵. Davis and coauthors [BDM83] showed that a slightly weaker version with 𝑐2 ‖𝐴 − 𝐵‖ instead of ‖𝐴 − 𝐵‖ on the AUGUST 2023 right-hand side of (3) is true. Later it was shown by Holbrook [Hol92] that 𝑐2 here cannot be replaced by 1 (as had been believed for years). In the four decades since the publication of [BDM83] no further progress has been made on this problem, nor any other method found to handle it with success. To complete this story, in another paper coauthored by Davis [BD84], it was shown that the inequality (3) does hold when both 𝐴 and 𝐵 are unitary. It was my good fortune that I met Chandler soon after my PhD and got introduced to these problems. Our collaboration began in 1980 and lasted until his death. As a collaborator he was generous and gracious. He was both intense and relaxed. After every discussion he typed the salient points and sent a note to his coworkers. But he would not hurry them on to publish. The three-author collaborations in [BDK89] and [BDM83] were coordinated by Chandler—I first met my coauthors McIntosh and Koosis much after the papers had been published. Chandler devoted a tremendous amount of energy to various progressive causes. Observing him, I was struck by two things. He always treated people with the opposite view with respect and patience, and no activity that could advance a good cause was too small for his attention. As he would say, losing one’s illusions does not mean one should lose one’s hopes. Man-Duen Choi The work of Chandler Davis influenced a large number of mathematicians. For more than 40 years, Chandler was the mainstay of the Toronto operator theory seminars that meet Monday afternoons. Here I will informally describe Chandler’s research interests in operator theory. Serious readers are referred to a longer article [CR94] with a large bibliography listing 80 papers written by Chandler. Chandler received his PhD from Harvard University in 1950. His doctoral thesis, written under the supervision of Garrett Birkhoff, was titled “Lattices of Modal Operators.” Birkhoff was famous for the book Lattice Theory and for developing connections with quantum mechanics. Three forerunners in operator theory had special impact on Chandler: • Mark Krein: Chandler could read Russian, a big advantage for operator theorists of Chandler’s age. Chandler’s early research showed broad interest in modern analysis as practiced by the Soviet school of Mark Krein, continued by David Milman, Mark Naimark, Israel Gohberg, Vadym Adamyan, Mikhail Livsic, and others. Man-Duen Choi is a professor emeritus of mathematics at the University of Toronto. His email address is choi@math.toronto.edu. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1111 • Bela Sz.-Nagy: Bela Sz.-Nagy was a leader of Hungarian school of analysis, in charge of the journal Acta Sci. Math (Szeged). Chandler’s favorite book was Analyse harmonique des opérateurs de l’espace de Hilbert [SNF67], coauthored by Sz.-Nagy and Foias. • Paul Halmos: Halmos was born in Hungary and had greatest impact in North America because of his excellent expository lectures and books. Indeed, Halmos’s book A Hilbert Space Problem Book [Hal67] has always been an inspiration for everybody in the field. Halmos visited Toronto in the 1970s. As highly motivated by Halmos, Chandler used 2-by-2 matrix techniques. Namely, the setting of a single operator 𝑇 in terms of four operators of the form 𝐴 𝐵 𝑇=( ) 𝐶 𝐷 is indispensable for Chandler’s major research articles concerning norm structure (e.g., joint work with Kahan and Weinberger [DKW82], compression problems, dilation problems in line with Sz.Nagy’s work, extension problems, and J-unitary structure as developed by Krein). Chandler introduced the notion of the shell of a Hilbertspace operator [Dav70]. This 3-dimensional analogue of the numerical range has intriguing relations with various geometric properties. Chandler also dealt with the interesting Toeplitz–Hausdorff theorem on numerical ranges [Dav71]. As generalization of the triangle inequality, the Cauchy– Schwarz-type inequalities related to different convex structures appeared often in Chandler’s research. In particular, a deep theory was developed for the Kantorovič inequality [Dav80], which is a useful tool in numerical analysis and statistics for establishing the rate of convergence of the method of steepest descent. Chandler claimed that two subspaces are easy, while three subspaces are much harder. This could be translated to fruitful results if a linear subspace was replaced by an orthogonal projection (as alias). In other words, the geometry of subspaces can be transformed to the algebraic features of the projections. Thus, the algebra generated by two projections is completely manageable, while the algebra generated by three projections becomes intractable. Chandler proceeded further to describe the angle between two subspaces, by means of the subtle cosine and sine functions. On the other hand, Chandler established the following two beautiful results in one of his earliest papers [Dav55]: • There exist three projections that do not have a nontrivial common invariant subspace. 1112 • The algebra of all bounded operators on a separable Hilbert space is generated (as a weakly closed self-adjoint algebra) by three projections. These two propositions are logically the same result connecting noncommutability with transitivity. Chandler’s research also concerned the effect of perturbations on eigenvectors of a Hermitian matrix. Related successful papers with Kahan, Bhatia, McIntosh, and others dealt with normal matrices. This topic is discussed by Rajendra Bhatia earlier in this memorial article. Aaron Davis I knew Chandler Davis as a father first and foremost, and as a man of faith. He was not religious but had a deep abiding faith in humanity. He was also driven by a love of life and an insatiable curiosity about it, whether looking through a scientific, mathematical, musical, literary, or poetic lens. Chandler’s American roots went back to the Mayflower and the early settlers of the Massachusetts Bay Colony, and also the first Quaker settlers of Pennsylvania. The tradition of political commitment ran deep on the abolitionist Quaker side of the family. Chandler’s great-grandfather Norwood Penrose Hallowell took a bullet as a captain in the Union army at the Battle of Antietam. The Hallowells ran a station of the Underground Railroad in Philadelphia. When my sisters and I were children, Chandler and Natalie would take us to civil rights and then anti-war demonstrations. After Chandler’s stand against HUAC, prison sentence, and subsequent blacklist, we moved to Canada, but he didn’t stop with political organizing, and we took in a succession of draft dodgers at our home in the late sixties. Although keenly analytical, Chandler was also a very expressive man, and his poetry conveyed his deep feelings. In his 1968 poem Toronto Home, Chandler wrote of what I remember as a tranquil scene: our study where we would do our homework around the fireplace while Natalie and Chan prepared their university classes and the cat purred. He saw that “we are not secure. . . ” and saw the cat as “timing our instability with neutral, implacable switching tail.” He saw the fire as casting “pseudopods that fade so fast” as the clock ticks. “We will not survive”, Chandler wrote, as our deaths are inevitable. But knowing that, he wrote: “Die with me, and in the waiting time, our life, caress and kiss, as if you meant to keep it, this temporary love. . . .” Even in his poems as a young man, Chandler dwelt on the impermanence of life but also of the longer arc of human generations and regeneration and of love and the fight for freedom being passed down somehow, of ideas Aaron Davis is a composer, arranger, and pianist living in Toronto, Canada. His email address is aaro@rogers.com. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 70, NUMBER 7 Figure 3. Chandler Davis and Natalie Zemon Davis, 2018. and life being reborn. He will be missed but left a legacy of hope in his words and deeds. Natalie Zemon Davis I had seventy-four years of marriage with my beloved partner Chandler Davis, and they were filled with conversation. Indeed, the first day we met—in our student days in 1948—we talked the night away in Harvard Yard. We talked of our hopes for the future—for ourselves and for the world. We talked of our intellectual interests and our literary favorites. But there was an asymmetry in our exchange. I could tell him all about the Senior thesis I was starting on at Smith College. It was about a Renaissance Aristotelian philosopher and Chandler could ask knowing questions about him. He could tell me about the poems and science-fiction stories he had been writing and I could follow along and comment with interest. When it came to his doctoral studies, however, I could make little headway. Yes, I’d heard of algebra and took a class in it in high school. But what was linear algebra? I asked for classes in the history of science at Smith, and then established a field in the history of early modern science for my graduate studies, but they didn’t leave me with anything pertinent to discuss about Hilbert spaces. The conversational asymmetry in regard to our work lasted throughout our marriage. Chandler patiently listened to or read my history manuscripts over the decades and made telling suggestions—for which I thanked him in my acknowledgments. I listened politely as he told me about a new idea or a new theorem, his eyes sparkling. I Natalie Zemon Davis is a historian of early modern times, the Henry Charles Lea Professor of history emeritus at Princeton University, and an adjunct professor of history at the University of Toronto. Her email address is nz.davis @utoronto.ca. AUGUST 2023 stood in admiration as he and other mathematicians exchanged information and made mathematical discoveries as they talked or reached for a piece of paper or a piece of chalk. Especially, I stood in wonder as conversation among mathematicians would suddenly stop and silence would prevail for several minutes. That would never happen with historians and anthropologists—we just went rushing on with our facts, ideas, and associations. Chandler and the other mathematicians would stop simply to think and patiently wait for each other’s conclusions. The other thing that struck me about Chandler’s scientific world was how international it was. I was used to have contact with historians from Europe who were interested in the same period and the same problems as I. But it was many years before I had links to scholars from Asia or North Africa—and then only because I had turned to problems in their regions. With Chandler, he was in communication with scholars from India and Japan already very early along. When I would see him and others at their international meetings, I was struck by how patient they were with linguistic barriers, how they did what was necessary to communicate. Chandler was deeply committed to helping mathematicians in different lands, including mathematicians suffering from political persecution. He also applauded the younger generations for their efforts and achievements. Even while pondering the philosophical question of uses of mathematics for society, Chandler never lost his delight in its beauty. Simone Weil Davis It’s a pleasure to pause and reflect on how it felt, as a daughter who did not go into math, to stand near to Chandler’s life as a mathematician. So much else of our parents’ pursuits were discussed and available to us. How strange it was, to be almost “black box” uncomprehending and yet to notice how this part of his life showed up in our home. From a kid’s point of view, at the height of a tabletop, I remember the abacus, its painted wood marbles impaled on parallel metal rods, all held forever in a wooden frame. With respect, I understood that this toy I played with was somehow also a tool, a way of figuring, a manifestation of questions and answers. The dreamily smooth slide rule pulled me in, too; at times as a grade schooler I would know, briefly, how to use it, due to Dad’s patient, inventive instruction. Simone Weil Davis teaches ethics, society, and law at the University of Toronto. Her email address is sdavis@trinity.utoronto.ca. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1113 Chandler brought that same creative, gentle practice to the math enrichment classes he helped lead on Saturday mornings; I was proud to see his clear-spoken encouragement of my classmates and friends. He believed in numeric ideas, and in the intellect of those he addressed (regardless of age). With a pencil and paper, he would let the logic of the numbers take us along. Beauty would follow, if we could stay upright in our mental kayaks long enough to appreciate it. In the early 1960s, he let me watch an animated short he was working on with a colleague in Madison, Max in Mathemagick Land, and quietly explained to me the rules of existence if there were to be a culture that lived in two dimensions. That was a burst of beauty. In her remarks above, Natalie mentioned Chandler’s many trips—sometimes extended journeys and often across the globe—to connect with his mathematical communities and project partners. As a child, while I’d miss him terribly, I also took note of the very particular kind of trust and interconnection, the bond, between him and his math-making colleagues. Most striking perhaps was his working friendship with Rajendra Bhatia over so many decades. We would pour over his letters from India, and I had a sense of him at ease in this other home, in this other land, climbing paths and walking under flowering trees that I’d never seen or breathed in. I would imagine him and Rajendra talking as they walked, or sitting in silence, jumping up to write on a chalkboard. It seemed such a peculiar, particular thing to me, his mathematical friendships. For one, it made the wide world appear traversable, and interconnection between people everywhere as natural, inevitable, even urgent. To an extent, the connections looked like an emotionally neutral partnership, shaped by rigor and large, shared puzzles. But against the backdrop of that cool profundity, everyone’s particular personalities stood out like wonderful, life-packed anomalies. Your Lee Lorch, your Peter Rosenthal, your Marjorie Senechal! Each person seemed outsized to me and with wonderfully interesting idiosyncrasies, extraordinary because of the bond they shared with my father. The largest invisible manifestation I sensed, when I looked at Chandler’s engagement with mathematics and his mathematical friends, was love. Love married to inquiry, logic married to imagination, and a bottomless curiosity, as shown in this excerpt from his poem Whether, which Chan was still revising just months before his passing. It could be someone’s looking at our world From distant worlds moving at near-light-speed, But is it meaningful to feel akin to them When we can’t invariantly distinguish whether They’re close in time or in our distant future? 1114 And yet I hope they watch and wonder, whether They think contemporaneously or not. If nothing in the cosmos tells us whether It opens out forever in space and time Or curls upon itself in space and time So that geodesics long enough are closed, How could it ever be conceived of whole? And yet I fondly love it whole, whether My concept of it has some truth or not. Hannah Davis Taïeb My earliest memories of my father as a mathematician come with a feeling of joy. My father sat on the floor with us when we were little, sharing ideas: matrices, sets, imaginary and real numbers, bases, different ways of looking at the finite and infinite, at the 4th dimension, at time. Somehow, the way he taught, it wasn’t “difficult;” he made it seem charmingly complex, delightfully intricate and precise, and therefore easy, pure, self-evident, each idea emerging from the previous one. His students, later, must have felt some of that same ease, I imagine. I remember doing the dishes with my father; I was nine or so; he challenged me to prove that 3, 5, 7 were the only examples of consecutive odd primes. I dried a few dishes; he had given me the tools to answer him, and so I could. Sometimes I say that “he taught us mathematics as children,” but it was more like a kind of play, as if his own pleasure at the way things fit together was just too much for him to keep to himself, and so it poured out on us. One biographer referred to my father as a “polymath.” I love this. His pleasure in abstraction—and in life—meant that he did not isolate his mathematical self, his mathematical reasoning, from all the other ways he had of engaging with the world. His science fiction writing, his poetry, his music—each domain of creation bordered on the others, and enriched the others. Going through his papers in recent weeks, I found clever skits about mathematical subjects, written when he was a student at Harvard. The link between poetic and mathematical expression carried throughout his life, culminating in the workshops on Creative Writing in Mathematics and Science at the math institute at Banff. The way he linked mathematics to every other aspect of his immense creativity made him a perfect person to be the editor-in-chief of the Mathematical Intelligencer. Hannah Davis Taïeb is the president of the nonprofit association Dialogue & Transformation. Her email address is dialoguetransformation@gmail .com. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 70, NUMBER 7 And, of course, his mathematical self also intersected with his politics. Chandler could have lived out his passion for justice in so many ways; he could have felt a conflict between his political ideals and his profession, Instead, he linked the two, by showing solidarity with mathematicians around the world, standing up for those with whom he shared a profession, a way of life. Along with my admiration for all that he has done is my pleasant memory of sitting on the floor as a tiny child, playing with numbers, colors, and ideas, and feeling that sharing of joy. John Friedlander I am honoured (okay, honored) to have been included amongst the invitees to contribute to this memorial article about H. Chandler Davis. Perhaps I did not know him as well as some of the other authors, but I did know him for a very long time and I do have some memories of him that I hope are worth recounting. As I recall, Chandler arrived at the University of Toronto and began teaching here sixty years ago, in the autumn of 1962. I took his third-year analysis course the very next year. It was a wonderful experience. Like almost all U. of T. math courses at the time, it was a full-year course. Unlike every one of the many other undergraduate math courses at the time (through all four undergraduate years), it met three hours per week whereas all the others met for only two. This of course was Chandler’s doing. In the Fall term we studied metric spaces, Banach spaces, then Hilbert spaces. I don’t recall any specific text though there were a number of them floating about. In the winter term the course covered an introduction to complex analysis using volume one of Hille’s two volume set. My preference for functions of a “complex” variable started right then. We had lots of talented students in the course, also before and after my year. Stephen Fienberg, who became a prominent statistician, took it in its first year (and let me hang onto his beautiful lecture notes during the following year). James Arthur took it the year after me. But there were several others who went on to distinguished careers. As the year progressed, we gradually picked up bits about Chandler’s history in the US that had taken place shortly prior to his coming to Canada. Being Canadian and being university-age students, you can guess where our sympathies lay, how we admired him for his courage. One day there was, amongst the students, a great buzz of excitement. It seems that one of the “guys” in the class had phoned Chandler at home, something unheard of in John Friedlander is University Professor of mathematics at the University of Toronto. His email address is frdlndr@math.utoronto.ca. AUGUST 2023 those days. He related the following: “A woman answered and said ‘Hello’.” There was a pause “May I please speak to Professor Davis?” Response without a pause “Which one?” I realize that these days such an exchange would create no excitement at all. But this was a very long time ago. We were all atwitter. It was quite a number of years before our paths crossed again. I returned to Toronto for good in 1980 and suddenly we were colleagues. Then, within a few years, I was Chandler’s chairman. This might have been a problem. I had always thought of Chandler as one who is tough on authority figures, although not on others, so I was a bit leery about being his chairman. But any genuine cause for this worry never did transpire. He was tenacious however. I remember being in Kyoto for the 1990 ICM and being engaged in mathematical conversation, catching up with my former postdoc Andrew Granville and being blissfully happy about it, when suddenly Chandler rushed up from out of a crowd, pointing his finger at me with the words “Some Departmental Business!” I was not kind: “Not here, not now!” There was one incident near the end of my term that I remember very well. It was then the law in the Province of Ontario, and most of the other provinces as well, that professors (and many others) faced mandatory retirement at the age of sixty-five. The Faculty Association and numerous individuals had been pushing hard for the abolition of this requirement for quite a number of years. So, Chandler came to my Chairman’s office one day and the conversation went almost verbatim like this. He began: I shall be sixty-five soon and I very much do not want to retire. So, I went to see the dean and he told me the following: “You do not need to retire. This is up to your chairman to decide. He can continue to pay your salary if he chooses to.” I should at this point state that I completely believed Chandler’s story. This dean was a person whose definition of the truth was “whatever you can get someone to believe”. So here is what I told Chandler: “On June 30 of the year your retirement is due, the Provost will take your salary out of the department budget. There is nothing that anybody can do about that. I have an annual departmental budget of some three to four million dollars. Out of that, I am unable to touch all but one account which has about one hundred and fifty thousand dollars. I use that account to help support our faculty in hiring postdoctoral fellows. I could give you that money but we would then not be getting postdoctoral fellows.” To Chandler’s credit, he both believed me (he too knew that dean) and he didn’t argue the point. This may sound obvious but some faculty have been willing to volunteer to NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1115 take that money. I can’t tell you how much I sympathised, then and now, with Chandler’s plight, partly because he proved over many years thereafter how valuable he still was to the community, partly because, due to the luck of having come along later, I narrowly escaped the same fate. In the following years my main contact with Chandler was electronic, refereeing the occasional paper for the Mathematical Intelligencer. He didn’t overburden me and I was always happy to hear from him. I want to close with a word about Chandler’s marriage. There was a period of close to twenty years when I was spending a lot of time travelling back and forth between Toronto and IAS Princeton. I cannot count the number of times that, when I did meet Chandler, it was in an airport, whether Toronto or Newark. No need to ask the reason for his trip. Every single time, it reminded me to reflect on what a remarkable love story these two people share! Mary Gray What more can be said about Chandler Davis, a hero to me and to many others from diverse backgrounds around the world? I can add that he was a feminist in the true sense of the word, in his life and work embodying a deep respect and appreciation for the aspirations and achievements of women and men, all the while campaigning in his multitalented way for us all to take responsibility to make a better world. And perhaps most important, Chandler provided the respect and support that bolstered our own confidence in ourselves, confidence and motivation to work on a multiplicity of human rights in the US and around the world. That Chandler remained steadfast in his determination, but with respect for the less enlightened, with his eyes on the prize—be the goal justice and equity for all or just moving along to the next step in the right direction—meant having a champion in our own community. We remember how when I and a handful of other women in mathematics came to recognize more than 50 years ago that while we faced obstacles we needed to organize to take responsibility for improvement, he was there, one of the first male members—and an active one—of the Association for Women in Mathematics, repeatedly challenging the establishment. When we hear talk about double-blind refereeing, I remember Chandler’s disdain for the prominent male mathematician who asked “How could we know that a paper was any good if we did not know who wrote it?” Mary Gray is Distinguished Professor of mathematics and statistics at American University. Her email address is mgray@american.edu. 1116 And when the AMS Council debated support for a young woman mathematician identified as among the “disappeared” in the dirty war in Argentina, Chandler did not join the skepticism about whether her mathematical output merited our support, a question not asked about the similarly situated male victims of human rights violations. At an early meeting of AWM at a JMM, an eminent mathematician allowed as how his university had once hired a woman whose research “turned out not so great” so they got rid of her; declaring he wasn’t about to hire a woman again, Chandler asked “Of how many male hires might you say the same?” When many mathematicians were demonstrating against the Vietnam War, Chandler not only went to Vietnam to meet with mathematicians in the North but succeeded in bringing recognition to a woman researcher who had taught mathematics for years in the midst of fighting for her country. When Chandler lost his case in the Supreme Court we knew that it was for the free speech rights of all of us— all genders, all colors. To have on our side someone, really an icon, who never gave up the struggle—a mathematician and a person who led a committed life in a way to which we might aspire—was an inspiration for which many will always be grateful. Stephen Halperin My connection with Mathematics at U of T began as a freshman in 1959 and continued for 40 years (with a break for graduate school at Cornell) until I moved to Maryland. So I never took a course from Chandler and I don’t think we met or knew each other until I returned as a young faculty member. Even then, we taught at different campuses and our mathematical interests were remote. And by the time I moved to the main campus, Chandler had retired. But I was a very interested member of the large audience in the spring of 1965 when Chandler hosted a teach-in (possibly the first at Toronto) on the war in Vietnam. He had invited speakers who were well known for their positions (both pro and anti the war), and was very clear that he expected an honest presentation, not a political rally. However, before he could open the event, a small group from the Berkeley Free Speech Movement appeared on the stage and announced that unless the distinguished pro-war professor was removed from the speaker list, they would shut down the teach-in. I will never forget Chandler’s response—he said: Stephen Halperin is a professor of mathematics at the University of Maryland. His email address is shalper@umd.edu. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 70, NUMBER 7 I spent time in a US prison because I would not provide the House Committee (HUAC) witch hunt with the names of people I knew. When you have done as much to protect free speech, then I will listen, but in the meantime just go and sit down. And they did. Chandler’s teach-in opened my eyes to responsibilities outside my discipline and really started me on my path to anti-war activism at Cornell. I may not have learned much math from Chandler, but his example of courageous integrity has stuck with me, and there have been times in my professional life when I have needed to remember it. Sheldon Axler One day in August 1983, I nervously walked several blocks in downtown Warsaw, carrying in my backpack several hundred leaflets produced by Polish mathematicians associated with the then-banned Polish union Solidarity (Solidarność). Martial law aimed at suppressing Solidarity had ended in Poland only one month earlier. Chandler Davis had asked me to deliver the leaflets I was carrying to a Polish mathematician who would arrange for their distribution to mathematicians attending the International Congress of Mathematicians, and of course I was happy to do so. Chandler’s involvement with the Polish mathematicians was typical of his lifelong support for human rights. I first heard of Chandler when I was an undergraduate, doing a summer research project on convexity. I came across Chandler’s paper on plane convex curves [Dav63a] that contains his now famous footnote about support from the Federal Prison System, as quoted earlier in this article by Peter Rosenthal. I was intrigued, and even more so after I learned about why the author was in prison. Then a bit later I read Chandler’s beautiful paper [Dav71] that explains why the numerical range of every operator on a Hilbert space is convex. Paul Halmos’s review of this paper in Mathematical Reviews refers to “the elegance of the proof” that Chandler had constructed. Finally I met Chandler in person when I was a graduate student at Berkeley. Chandler’s wife Natalie, an eminent historian of early modern France, was a history professor at Berkeley at that time, so Chandler often visited Berkeley from Toronto. We had many conversations, and I came to greatly admire Chandler’s insight about multiple subjects, both mathematical and nonmathematical. Sheldon Axler is a professor emeritus of mathematics at San Francisco State University. His email address is axler@sfsu.edu. AUGUST 2023 In 1987, I became editor-in-chief of the Mathematical Intelligencer. The Reviews section of the Mathematical Intelligencer was responsible for publishing a few reviews in each issue, mostly of books that would interest mathematicians but also of other relevant items such as mathematical art. Because of Chandler’s deep knowledge of so many aspects of the history and culture of mathematics, I invited him to be the reviews editor of the Mathematical Intelligencer. I was delighted when he accepted this invitation. Of course Chandler did a terrific job as reviews editor of the Mathematical Intelligencer. This was the time when Chandler and I started the custom of having dinner together, just the two of us, one night each year at JMM to discuss multiple topics of mutual interest. When my term as editor-in-chief of the Mathematical Intelligencer entered its final year, I was happily surprised when Chandler was willing to become the next editor-inchief. I think he had a lot of fun with his magnificent handling of the Mathematical Intelligencer. At about the same time, Chandler was elected vice president of the American Mathematical Society, quite a reversal from the McCarthy period when he was blacklisted by American universities. The AMS had behaved decently during part of that time, hiring Chandler as an associate editor at Mathematical Reviews at a time when American universities were too frightened to offer him employment. Chandler leaves a huge legacy as the model for someone who does the right thing in difficult circumstances, while continuing to make important contributions to mathematics. Marjorie Senechal It was a great privilege, and a great pleasure, to work closely with Chandler Davis for many years. He was an outstanding mathematician, a committed activist, a celebrated fiction and science fiction writer, a fine poet, a composer of gracious art songs, and the visionary editor of the international quarterly journal, The Mathematical Intelligencer. Chandler’s many brilliant facets reflected a single sui generis whole. This unity was especially evident in the week-long creative writing workshops in mathematics and science that he and I co-organized at the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Canada in 2003, 2004, and 2006 (the last together with the poet and philosopher Jan Zwicky). You can see it in our workshop anthology, The Shape of Content. Marjorie Senechal is Louise Wolff Kahn Professor Emerita of mathematics and of history of science and technology at Smith College. Her email address is senechal@smith.edu. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1117 Figure 4. Chandler’s books include The Shape of Content, an anthology of writings from the BIRS creative writing workshops. We organized the workshops to encourage practitioners who engage this content in their work, to give them opportunities to discuss important issues, to learn what others are doing, to encourage each other, to critique current work, to welcome young writers into the field, to spark collaborations, and to forge networks and build community. In that sense, the creative writing workshops’ goals were the same as any other BIRS workshop’s. But we based these workshops on the premise that Chandler himself personified: mathematics and science are part of world culture, part of the human spirit and, as such, are fitting subjects and themes for poetry, drama, short stories, novels, nonfiction, comic books, essays, film, and even music. As we noted in our workshop proposals, creative writers don’t coerce their audiences to eat mathematics and science like medicine hidden in jam, they convey these ideas through art instead of formalism. True, plays like Proof and biographies like A Beautiful Mind and The Man Who Loved Only Numbers might have been less successful had the mathematician character been less idiosyncratic, but the play Copenhagen was also a great success. The novel Einstein’s Dreams conveys the scientific creative process in a beautiful way and Arcadia, a funny and chaotic play whose leitmotif is chaos theory, is a modern classic, and the mathematical formalism is symbolized in its structure. Chandler’s egalitarian spirit infused the workshops. There were no leaders: everyone learned from everyone else, mathematicians and non-mathematicians alike. And creative writing was sparked by cross-genre insights: a poet helped a fiction writer find a better way to end his story, a mathematician nonfiction writer helped a dramatist extend the ideas of her play, ideas a filmmaker sitting in on 1118 their discussions recast in doggerel form. A novelist had insightful comments on poetry. As Chandler explained ([Dav08]), I remember Norberto hadn’t brought his white cane so going out for coffee on unfamiliar streets he gladly held for guidance my gladly offered elbow. And in mathematics, it was the blind leading the blind! Whenever one of us had guidance to give the other, it was a gladness to be giver, it was gladness to be receiver. ”I see, I see,” Norberto murmured. References [BD84] Rajendra Bhatia and Chandler Davis, A bound for the spectral variation of a unitary operator, Linear and Multilinear Algebra 15 (1984), no. 1, 71–76. MR731677 [BDK89] Rajendra Bhatia, Chandler Davis, and Paul Koosis, An extremal problem in Fourier analysis with applications to operator theory, J. Funct. Anal. 82 (1989), no. 1, 138–150. MR976316 [BDM83] Rajendra Bhatia, Chandler Davis, and Alan McIntosh, Perturbation of spectral subspaces and solution of linear operator equations, Linear Algebra Appl. 52/53 (1983), 45– 67. MR709344 [BR97] Rajendra Bhatia and Peter Rosenthal, How and why to solve the operator equation 𝐴𝑋 −𝑋𝐵 = 𝑌 , Bull. London Math. Soc. 29 (1997), no. 1, 1–21. MR1416400 [CR94] Man-Duen Choi and Peter Rosenthal, A survey of Chandler Davis, Linear Algebra Appl. 208/209 (1994), 3– 18. MR1287335 [Dav55] Chandler Davis, Generators of the ring of bounded operators, Proc. Amer. Math. Soc. 6 (1955), 970–972. MR73138 [Dav63a] Chandler Davis, An extremal problem for plane convex curves, Proc. Sympos. Pure Math., Vol. VII, 1963, pp. 181– 185. MR0154189 [Dav63b] Chandler Davis, The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6 (1963), 159–173. MR149309 [Dav70] Chandler Davis, The shell of a Hilbert-space operator. II, Acta Sci. Math. (Szeged) 31 (1970), 301–318. MR273447 [Dav71] Chandler Davis, The Toeplitz-Hausdorff theorem explained, Canad. Math. Bull. 14 (1971), 245–246. MR312288 [Dav80] Chandler Davis, Extending the Kantorovič inequality to normal matrices, Linear Algebra Appl. 31 (1980), 173–177. MR570389 [Dav88] Chandler Davis, The purge, A century of mathematics in America, Part I, 1988, pp. 413–428. MR1003186 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 70, NUMBER 7 [Dav08] Chandler Davis, Guided (Chandler Davis, Marjorie Wikler Senechal, and Jan Zwicky, eds.), A K Peters, Ltd., Wellesley, MA, 2008. Creative writing in mathematics and science. MR2516493 [DK70] Chandler Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46. MR264450 [DKW82] Chandler Davis, W. M. Kahan, and H. F. Weinberger, Norm-preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal. 19 (1982), no. 3, 445–469. MR656462 [Hal67] Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR0208368 [Hol92] John A. Holbrook, Spectral variation of normal matrices, Linear Algebra Appl. 174 (1992), 131–141. MR1176456 [HB93] Lars Hörmander and Bo Bernhardsson, An extension of Bohr’s inequality, Boundary value problems for partial differential equations and applications, 1993, pp. 179–194. MR1260445 [Sch86] Ellen W. Schrecker, No ivory tower: McCarthyism & universities, Oxford University Press, New York, 1986. [SN53] Béla Sz.-Nagy, Über die Ungleichung von H. Bohr, Math. Nachr. 9 (1953), 255–259. MR54765 [SNF67] Béla Sz.-Nagy and Ciprian Foiaş, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson et Cie, Paris; Akadémiai Kiadó, Budapest, 1967. MR0225183 EARLY BIRD SPECIAL ON AMS MEMBERSHIP! What does AMS membership mean to you? Networking and professional development Community and camaraderie Supporting the mathematical sciences Credits Figure 1 is courtesy of Johnny Siera. Figures 2 and 3 are courtesy of Natalie Davis. Figure 4 is courtesy of Marjorie Senechal. Renew or Join by Oct 1, 2023, and get AMS Points*! AMS members: Renew for 2024 by October 1, 2023, and receive 15 AMS Points. Become a Life member and receive 30 AMS Points! New or Lapsed Members: Join by October 1, 2023, and receive 15 AMS Points. *See www.ams.org/amspoints for full details. Exclusions apply. www.ams.org/membership AUGUST 2023 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1119