Explore the groundbreaking contributions of Evelyn Boyd Granville, the second Black woman to earn a Ph.D. in mathematics in the U.S., whose pioneering programming work supported key space projects like Project Vanguard and Project Mercury. This article sheds light on her vital role in early space computing, bridging academia, government, and industry.
Article By Laura E. Turner (Originally published by the American Mathematical Society)
The second Black woman to earn a PhD in mathematics from an American university, Evelyn Boyd Granville (1924–2023) had a career that spanned academia, government, and industry, and included groundbreaking work in computing in the prehistory and early years of the US space program. Since most accounts of her life provide little information about the latter facets of her work, I aim to elaborate upon [Tur23] by shedding light on the sorts of programming in which Granville and her colleagues were engaged.
In what follows, I will first treat aspects of the early stages of space computing, with particular emphasis on the projects in which Granville was involved. Here, the primary focus is IBM work in relation to Project Vanguard and Project Mercury, though a limited treatment of other projects on which Granville worked, both at IBM and elsewhere, is included. I will then describe some of the roles Granville played in these connections and, when possible, some details of her individual involvement. Because her name rarely appears in scientific publications, it is difficult to determine her specific contributions to these projects; future work remains to uncover this information. In light of this, my foremost aim is to provide a new context in which to appreciate and understand this pioneering work.
International Business Machines (IBM) took its name in 1924. Founded in 1911 through the merger of three manufacturing businesses, the company initially sold punch card tabulating equipment and a variety of other products ranging from butcher scales and coffee grinders to time clocks and office furniture [Cor18, p. 121]. Although the demand for advanced office machinery increased during World War I and through the 1920s, the company faced challenges linked to patent issues, competition, and the increasing complexity of its machines. It reaped the benefits of its “optimistic strategy” for product development, however, in a lucrative contract with the US Social Security Administration for tabulating equipment. This set it on a path for growth even during the Great Depression. It shifted from tabulating equipment to computers during the late 1940s and into the 1950s, a transition supported by US government agencies. With a ready commercial market for computing products and a role as an important provider, IBM expanded dramatically. In 1950, its worldwide workforce numbered just over 30,000. By 1963, this number had more than quadrupled [Cor18, pp. 132– 133] 25 man-years to the project and wrote close to 50,000 instructions for the 704 and 709 [Mow60, pp. 120–126]. Predicting the course of a satellite was challenging due to its tremendous speed. Once a satellite was launched from the Cape Canaveral Missile Test Center, the VCC was tasked with calculating its orbit using data from the Minitrack stations around the world. Once a station received a signal, it sent a message in triplicate to the Vanguard Control Center at NRL for review. The information was then relayed, again in triplicate, by teletype (a precursor of the fax machine) to the VCC, where it was received on punched paper tape. There, an operator fed the tape into a machine that punched holes in IBM cards corresponding to the Minitrack observations. After these were fed into a card-reader that transferred the information to the 704, a master program processed the observations, calculating the predicted longitude and latitude of the satellite positions for each minute, one week to ten days in advance. Predictions could be entered on punch cards, converted to teletype, and sent back to the Control Center and Minitrack stations, providing advance information to observers concerning the time and angle at which to expect the satellite for the purpose of optical location. Output devices also included a cathode-ray tube visual display and a cathode-ray tube recorder [Hag58, pp. 48–50].
As the company grew, its tradition of equal employment opportunities became a point of pride. A 1953 employment policy letter states: “It is the policy of this organization to hire people who have the responsibility, talent and background necessary to fill a given job, regardless of race, color or creed” [IBM65]. A 1965 issue of IBM News indicated that the company had been actively recruiting Black employees [IBM65], and a 1957 brochure for women (titled “My Fair Ladies” after the recent hit on Broadway) encouraged women applicants for positions in computing. During the same period of growth, IBM also won government contracts for work on important new projects, pushing the boundaries of computing at that time. The US satellite program, called Project Vanguard, was one of them.
The period from July 1, 1957 to December 31, 1958 marked the International Geophysical Year (IGY), an international program of geophysical research initiated in the midst of the Cold War. Sixty-seven countries, including the United States and the Soviet Union, participated in this effort toward a comprehensive study of the earth’s atmosphere, surface, and interior, as well the measurement of nuclear radiation on land and in the air and sea.
Project Vanguard was one part of the US Scientific Satellite Program connected to IGY. Initiated in 1955, it aimed at launching a scientific satellite, to be designed and built by the US Naval Research Laboratory (NRL), during that 18-month period. The objectives of the project were threefold: to put a satellite into orbit about the earth, to prove that it was in orbit, and to conduct at least one scientific experiment using it. One requirement for this earth satellite program was a high-speed digital computer for calculating a satellite’s orbit. By 1956, the Office of Naval Research, which oversees NRL, invited bids from companies equipped to rent computer facilities and provide mathematical and programming services, and IBM won. For $900,000 (roughly $10,000,000 in 2023), IBM would supply its 704 computer for six weeks, plus orbital computations for the entire lifetime of the satellite or the Minitrack tracking system, whichever was shorter, for the first three successful satellites to be launched. In addition, it would furnish the services of mathematicians for programming (designing and creating programs), coding (writing programs in machine language), numerical analysis and related tasks, as well as 100 hours of computing time for checking programs, and a computing center in DC to be made available on demand [GM70, p. 160]. A. Robin Mowlem, who directed programming there, described the NRL decision to enter into this contract as “a leap into the future,” for “at that time one could not be sure that satellites would behave exactly the way they have or, indeed, that computers were suitable for orbital computation” [Mow60, p. 119].
The Vanguard Computing Center (VCC), as it would be called, opened in July 1957 in what was once a bus terminal. Its 704, a state-of-the-art computer with vacuum tube logic circuitry and magnetic-core memory capable of storing up to 32,768 words, was designed with scientific, industrial, business, and government calculations in mind; much faster than earlier models, its floating-point arithmetic hardware made it ideal for precise mathematical computations such as those involved in Project Vanguard.
Proof that a satellite was in orbit necessitated tracking it, computing its orbit, and establishing a predicted path over the earth’s surface. Tracking Vanguard satellites involved radio angle and optical methods, the latter of which were important in making precise measurements of the position of the satellite, and thereby obtaining the best determination of its orbit. These satellites had no selfcontained light source, but if one crossed the twilight belt, it could be viewed, if large enough, by the naked eye or through binoculars or telescopes by observers at that latitude [Hag58, p. 36]. The VCC was primarily concerned with processing positional information obtained through radio transmission via the Minitrack tracking system developed at NRL. Vanguard satellites were equipped with 108Mc signal sources, with Minitrack stations positioned in a worldwide network. When a station received a satellite transmission, it was subdivided into individual observations consisting of phase-difference readings for the northsouth and east-west antennas of that station [QJ58, p. 59]. The Minitrack system functioned as a radio interferometer calibrated against the stars, providing direction cosines of the position vector of a satellite (the cosines of the angles between the vector and the coordinate axes) relative to the tracking stations. Once transmitted to the Control Center, and then to the VCC, the information was ready to be processed by the 704.
Work on the orbit computation methods to be used began immediately, and once the algorithms were formulated mathematically, mathematicians and programmers at IBM translated them into programs, first to be run on the 704, and later the 709, a faster and more powerful vacuum tube successor. Elliptic or circular orbit programs were used to determine the main features of the orbit, and precise orbit programs were then used to find the precise orbit. There were two types of each program: for the former, it depended on whether or not the observations belonged to the same neighborhood in the sense of a Taylor series, and for the latter, one used numerical integration and the other the method of general perturbations (to be touched upon later) [Sir58].
Programming, which was generally machine symbolic (FORTRAN was incomplete when the project began, but was increasingly used later), was a task of enormous magnitude. Mowlem estimated that IBM devoted roughly
The VCC was responsible for satellite tracking and orbit determination, with IBM staff handling the programming. The system was highly automatic, with minor decisions governed by the algorithms, and based on collections of subroutines linked into macro-operations; this minimized duplication, allowed for easy modifications, and enabled programmers to work independently on the various parts of the system [Mow60, p. 126]. It was also card-controlled for greater flexibility. Upon receiving a transmission, subroutines loaded the message into the high-speed storage of the 704; compared the triplicate items for agreement; adjusted the data, converting phase readings into directional information; and fit a least-squares parabola to the direction components from the phase-difference readings. These were linked into the initial macro-operation, which produced a “smoothed” direction of the satellite from that Minitrack station for one instant of time, expressed according to a local coordinate system [QJ58, p. 60].
Another macro-operation used one of several different procedures based on Gauss’s method to compute a preliminary orbit and then improve it iteratively. Next, the approximate orbit was refined through numerical integration of the differential equations of motion relating (by Newton’s law) the components of the gravitational and atmospheric drag forces on the satellite to its acceleration components, providing the predicted position and velocity vectors for the satellite across time. A different macro-operation performed a differential correction method to the position and velocity vectors to bring predictions into closer agreement with observations. Separate macro-operations then “translated” the predicted position, height, and zenith-angle-acquisition for a specific time, latitude, and longitude into a form which was more meaningful to the general public.
An alternate technique for obtaining predicted positions was implemented by 1960 via a general oblateness perturbation program based on Hansen’s lunar theory,[1]which computed Fourier series representations of orbital characteristics like perturbations of the satellite due to the oblateness of the earth (which has an equatorial bulge) in terms of time, and a separate numerical integration technique treating drag perturbations [QJ58, p. 61]. One key advantage of this program was that it could be evaluated at any particular time in a satellite’s orbit, which prevents the possibility of cumulative errors. This process, together with differential correction, was used iteratively to minimize the corrections to the orbital elements, and because of its long-term accuracy, was instrumental in the determination, by NASA scientists, that the earth is actually slightly pear-shaped [Mow60, pp. 124–125]. In fact, in their report of this discovery in Science, John O’Keefe, Ann Eckels, and Ken Squires thanked “the Vanguard Minitrack Branch, the IBM Vanguard Computing Center [emphasis added], and Dr. Paul Herget, whose work in obtaining and processing the data made this study possible” [OES59, p. 566].
IBM staff began programming the 704 in 1956. When Sputnik I was launched in October 1957, their programs successfully computed its orbit using three satellite positions and, according to Donald A. Quarles, Jr., IBM’s chief mathematician for the project, when Vanguard 1 was launched the following spring, the VCC predictions were “accurate to within a small fraction of a minute of time ” [QJ58, p. 64]. Although the project experienced delays and launching and orbital failures, Project Vanguard met its scientific objectives. Three satellites were placed in orbit, and the data received by Vanguard 1, alone, established new findings about the shape of the earth; that the sun and the moon modified the orbit of earth satellites; that the orbital path of a satellite is affected by solar radiation pressure; and that magnetic drag dampens a metallic satellite’s rotational movement. As the orb continued its passage through space, it also provided information about the diameters of the earth’s equator and poles; the density of the upper atmosphere; and variations in atmospheric density with the rotation of the sun [GM70, p. 244].
Project Mercury was the first phase of the US manned satellite program. Its aims were to launch a manned satellite into orbit about the earth; recover the astronaut-pilot upon its return; and enable the study of human capabilities when subjected to the stresses of acceleration, weightlessness, deceleration, and landing. Alan Shepard became the first American in space under Project Mercury in May 1961, and IBM computers and staff were critical to this achievement, as well.
Although Western Electric won the contract for equipping and testing the Project Mercury Tracking and Ground Instrumentation System, based on its previous success in determining satellite orbits in Project Vanguard, IBM was awarded the subcontract for computers and software (this was later broadened to include designing and installing the Launch Monitor Subsystem). By then, the VCC had been renamed the IBM Space Computing Center (SCC), and featured the IBM 7090, a transistorized system 7.5 and six times faster, respectively, than the 704 and 709 (but far less powerful than a modern smartphone). The National Aeronautics and Space Administration (NASA) was founded in 1958, and when it opened the Goddard Space Flight Center in Greenbelt, Maryland, in 1959, its Computing and Communications Center housed duplexed 7090s, redundancies receiving the same input data and performing the same computations. They also served as the communications link between the Control Center and the remote radar stations, and were complemented by a 709 in Bermuda.
The 7090s were responsible for providing powered flight trajectory parameters (to be monitored for signs of a possible imminent abort) and a smoothed present position for Mercury capsules during the launch phase; predicting future positions and providing radar acquisition data; quantitatively monitoring radar and computer performance; and calculating and transmitting display information to the Control Center, with output data updated every half second during launch and abort, and 10 to 20 times per minute in other phases [Gas99, p. 42]. Programming for the different phases was integrated into a single automatically-sequenced package. Once the liftoff signal was received, programs performing launch computations were activated [Gas61, p. 40].
One role of the Goddard duplex during the launch phase was to make a “go” or “no-go” recommendation depending on whether the satellite had successfully attained an orbit. Doing so required computing the anticipated orbit lifetime and providing trajectory parameters needed for monitoring the launch status for any indications of the possible need to abort. Once the computers recommended a “go” decision and the flight controller had no external reason to recommend otherwise, the launch switch signaled the computers to execute the orbit program.
During the orbit phase, the 7090s calculated precisely when to fire the retro-rockets for reentry and landing. These times were recomputed after each new set of orbit parameters was determined, and position and time data were sent to Goddard from radar sites around the world. These data were used to predict the position of the capsule for acquisition and supervision by radar sites using methods like Cowell’s numerical integration to extrapolate and correct orbital parameters. Because its accuracy depended on the accuracy of the initial position and velocity vectors and the accuracy of the approximated perturbations due to the shape of the earth and its atmosphere, differential correction was used to improve the predictions [Gas61, pp. 40–41].
For reentry, the primary task was computing retrofire clock information and the capsule’s probable impact point (which was refined upon the arrival of new radar observations) using position and time data from the radar stations, and transmitting this information to the Control Center. Reentry trajectory data were then provided to radar stations in advance of the satellite’s passage [Gas61, p. 41].
Project Mercury was computationally demanding, and the significance of the achievement may be difficult for modern readers to appreciate. Real-time computing was incredibly rare at that time, and the sorts of programming methods required for the task of human spaceflight were previously nonexistent. Saul Gass, who managed the IBM Project Mercury Simulation Group in 1960, noted in a later description of Project Mercury’s computer system that his personal computer in 1999 had “greater computational ability and memory than all the combined computers used in a Project Mercury mission” [Gas99].
Aerospace Computing Beyond Vanguard and Mercury IBM was not the only US corporation engaged with orbit computations in the early years of the space age, nor were Vanguard and Mercury the only projects. Space Technology Laboratories (STL), for example, formerly the Guided Missiles Research Division of Ramo-Wooldridge Corporation (a private missile research firm), was then developing missile systems and spacecrafts and employed mathematicians, programmers, scientists, and engineers in this connection. In 1957, STL had developed a two-stage Advanced Reentry Test Vehicle (ARTV) for the US Air Force Ballistic Missile Division (AFBMD) that combined a Thor ballistic missile with the second stage propulsion system (the Able rocket stage) developed for Project Vanguard. This suggested the possibility of placing a probe in lunar orbit, and thus STL became involved in the first US effort toward a lunar mission; another part of IGY, this probe was intended to carry a camera and other scientific instruments.
Under AFBMD and NASA, STL was assigned important technical tasks which included not only engineering challenges but also computing and data processing to locate the payload in the event of a successful launch. Its Systems Research and Analysis Division was involved with managing space and missile systems studies and contained its Computation and Data Reduction Center (CDRC), which housed state-of-the-art IBM computers. There, staff performed functions including numerical analysis, applied mathematics, statistical analysis, scientific and computational systems programming, data processing analysis, and test evaluation programming and analysis.
North American Aviation (NAA) was another company active in the booming aerospace field. Incorporated in 1928 as a holding company, NAA developed as an aircraft manufacturer in the 1930s, and in the 1950s built aircraft for the US military and NASA for research on flight conditions beyond the earth’s atmosphere. By 1961, its newly developed Space and Information Systems Division (formerly its Missile Division) sought to accelerate the development of the NAA Hound Dog supersonic cruise missile, and engage in anti-ICBM (intercontinental ballistic missile) projects, manned and unmanned space exploration vehicles, and managing information processing systems. Notably, in 1961 it was also chosen as the prime NASA contractor to design, develop, and construct the spacecraft command and service modules for Project Apollo, which landed humans (Neil Armstrong and Buzz Aldrin) on the moon for the very first time in 1969.
IBM, too, took on new projects during the same period and in the years that followed. As one example, a team of its mathematicians was engaged in work on the US Air Force’s Athena spacecraft reentry research and development program at White Sands Missile Range. The Athena RTV (Reentry Test Vehicle) missile was developed within the Air Force Advanced Ballistic Missile Re-Entry System (ABRES) program, with the intention of improving ballistic penetration via smaller and less-expensive missiles than those launched from Cape Kennedy and Vandenberg Air Force Base. IBM was also involved in Project Gemini and Project Apollo, providing mainframes, software, and technical support, as well as Skylab (the first US space station) and the US Space Shuttle program.
Granville was involved in many of the projects described above as a mathematician and computer programmer. She was hired at IBM in 1956, having already distinguished herself academically and professionally. Born in DC, she attended segregated public schools with excellent teachers, and excelled at Smith College, graduating summa cum laude in 1945. She earned her doctorate from Yale in functional analysis in 1949 under the supervision of Einar Hille, and subsequently worked as a research assistant at NYU. She was a mathematics professor at Fisk University, where she taught Etta Zuber Falconer and Vivienne Malone Mayes,[2]from 1950 to 1952, and then returned to DC to take a position at the National Bureau of Standards (now the National Institute of Standards and Technology). By her own testament, this was where she met mathematicians working as programmers and contemplated a career in that field [Gra89].
When Granville (then Boyd) arrived at IBM, she had no experience with computers (this was not unusual), and attended a two-week training session at the Watson Computing Center in New York City. There, she became acquainted with the IBM 650 Magnetic Drum DataProcessing Machine, first introduced two years prior, and the Symbolic Optimal Assembly Program (SOAP) assembler language. She spent the next year in the IBM DC office writing programs [Gra89], though her activities were at least occasionally more varied. A 1957 feature in an IBM magazine describes how “Dr. Evelyn Boyd, assisted by Mr. [Saul] Gass, demonstrated basic Electronic Accounting Machines and the [IBM] 650 computer” to high school students during a seminar “designed to stimulate interest among high-school students in the new and important professions associated with the ever-expanding field of electronic computers” [IBM57].
Gass (the same man who would later manage the IBM Project Mercury Simulation Group), was a former Pentagon employee involved in scientific computation who left government when the Eisenhower administration cut funding for such work. In 1957, he was employed at IBM as an applied science representative, that is, as a technically trained professional who helped salespeople with customers. In this role, he described himself as an “...educator, as well as a pseudo-salesman,” and it was in this capacity that the local IBM applied science staff organized the seminar, busing 500 high school students from DC and nearby Maryland to the Mayflower Hotel [Gas99, pp. 38–39]. There, students had the opportunity to use the IBM 650 to evaluate a square root, with one student apparently remarking: “much better than a slide rule” [IBM57].
Granville moved to New York City as a research mathematician at the New York Data Processing Center of the Service Bureau Corporation, an IBM subsidiary which provided electric data processing services to customers. After IBM opened the VCC in Washington, DC, she transferred there, arriving around 1958. In doing so, she became involved with Project Vanguard.
Although her name does not appear in publications detailing IBM work on the project, Granville’s re´sume´ indicates she was engaged in “Computer programming” for the 704, and the “Formulation of orbit computations and computer procedures for Project Vanguard and Project Mercury.”[3] She marveled in a later interview that her work involved “writing programs for something up in the air the size of a grapefruit!” [Lam14].
This “grapefruit” was Vanguard 1 (officially 1958 Beta 2), launched on March 17, 1958. Vanguard 2(E), the first Vanguard satellite placed in orbit under NASA and a precursor to modern weather satellites, was launched in February 1959. It contained photocells to scan the earth and map its cloud cover. A wobble in its orientation made the interpretation of the data difficult, but it proved the feasibility of a weather satellite. With a 20-inch diameter, it was not the diminutive sphere Granville recounted in her interview, but she worked on this project, too, with her role as a “mathematician programmer” at the VCC highlighted by the Associated Press [Pre59].
Granville was also involved with Project Mercury, which began in May 1958 and concluded in May 1963. She left IBM in 1960 when she married Reverend G. Mansfield Collins and moved to Los Angeles. By that point, she served as an IBM Staff Assistant helping to solving trajectory problems [IBM66]. According to Ebony Magazine, which published a short profile of the “space computing mathematician,” she supervised the work of three people developing the calculations for tracking capsules [Mag60]. Attesting to the cutting-edge nature of this activity, in 1966 Granville indicated that her work on Project Mercury had involved “a constant learning process due to the ever improving mathematical techniques for trajectory calculations” and afforded her “the opportunity to stay abreast of the latest developments in the aerospace field.” She described this work as “most exciting because of the experience we had in formulating and implementing the calculations for the Mercury flights”—“one of the highlights” of her career with IBM [IBM66].
After she left IBM, Granville (now Collins) resumed work on the west coast. Since, she later recalled, IBM had no big projects in California in 1960, she was not able to transfer. Instead, she joined the technical staff of STL in Redondo Beach. There, she worked in its CDRC (Computation and Data Reduction Center) on “research studies on methods of orbit computations” [Gra89].
The CDRC contained an Applied Mathematics Department (AMD). STL was then under contract with NASA, and engaged in studies of earth satellite orbit computations and the effect of atmospheric density thereon. In this connection, AMD staff produced a final contractor report treating the Diliberto general perturbation method. In astronomy, perturbation refers to the deviation in motion or orbit of a celestial body from its trajectory due to forces such as (in the case of near-earth satellites) drag and the earth’s oblateness. General perturbation methods, important in the context of long-term motion, were analytical procedures used to express the deviations of a body from its unperturbed orbit, and allowed for the calculation of the perturbed position of the body at any point in time. By 1962, a number of general perturbation methods had been proposed. These included Hansen’s method, successfully used by IBM in Project Vanguard, and an adaptation of Delaunay’s method. The Diliberto method was a novel approach developed at STL based on Stephen P. Diliberto’s theory of periodic -surfaces ( -dimensional toruses composed of trajectories). In the AMD Report, the method was applied in a “new and more suitable” coordinate system, simplifying the analysis and results and, by introducing a change of variables, removing low eccentricity singularities. In addition to Diliberto, the five contributors to the report included E.B. Collins [STL62].
Granville left STL in 1962. By then, the American space program was dominated by manned spaceflight, first under Project Mercury, and then Project Apollo. Granville was involved in this latter project, too, at NAA, having been enticed there by a friend in August 1962. “It sounds like job jumping,” she later remarked, “but that was the way things were then. The whole field was exploding, and people needed workers. I was always moving on to more money and more interesting work” [Lam14, p. 26].
At NAA, Granville became a research specialist, giving technical support related to celestial mechanics, trajectory and orbit computations, numerical analysis, and digital computing to Apollo engineering departments [Gra89]. She left NAA in October 1963, however, after a call from IBM convinced her to return [Lam14]. She accepted a position in Los Angeles as Senior Mathematician in the Systems Development West Department of the Federal Systems Center (FSC) [IBM65], part of the IBM Federal Systems Division (FSD). There, she provided technical support in trajectory analysis, orbit computation, numerical analysis, and digital computer methods. In particular, she was engaged in data processing in the context of ballistic missile reentry and target recognition programs [IBM66] as a member of the IBM team of mathematicians working on the US Air Force Athena spacecraft reentry research and development program [IBM65]. By 1966, Granville worked in the IBM FSC-West Coast Operations’ Signal Analysis Department on a project for the space exploration program of the Jet Propulsion Laboratory treating the changes in velocity induced by the collision of particles [IBM66]. According to her own account in [Gra89], her work at the IBM FSD was “similar to that done at NAA—trajectory analysis and orbit computation using techniques of numerical analysis.”
When IBM reduced its staff in the L.A. area, Granville took a job at California State University, Los Angeles, in 1967 rather than transfer back to DC or elsewhere in the state. She and Collins divorced the same year. She married Edward Granville in 1970 and retired in 1984, at which point the couple moved to East Texas. Granville reentered the workforce there several times to teach at the junior high, high school, and university levels, and retired once and for all in 1997 [Lam14].
When asked, in 2015, about her “favorite job” across the span of her career, Granville cited her work related to the space program, referring to it as “something brand new.” “NASA was new,” she remarked. “And the idea of being able to write programs to track these satellites. That was really the most fascinating job I’ve had, yes” [PG15].
Granville’s work in the aerospace industry is now widely recognized and valued. Few accounts of the nature of this work and its broader context, however, have thus far accompanied descriptions of her life and trajectory, particularly in popular literature. While work remains to ascertain and explain her individual contributions to the US space program, it is the hope that the present account further illuminates and contextualizes this fascinating facet of her career, and the significance of the work of Granville and her colleagues at the vanguard of space computing.
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This article was originally published on the American Mathematical Society's website on December 31, 2024. It is reprinted here with permission. The content remains unchanged except for formatting adjustments for this website.
Citation: First published in Notices Amer. Math. Soc. 71 (December 2024), published by the American Mathematical Society. ©2024 American Mathematical Society.
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