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Biography of George E. Andrews
In high school, mathematician George E. Andrews was told by a guidance counselor that it was impossible to find a truly interesting career, so he should find something dull but practical to study. It turned out to be great advice for Andrews to ignore. “I chose to do what I loved,” he says. And the mathematical puzzles he grew to love possessed surprising links to ideas throughout 20thcentury science and mathematics. Andrews has followed his mathematical inspirations through a career that has taken him from the splendor of a footnoted formula to the unearthing of a mathematical treasure in a university library in Cambridge, England. Along the way, he has solved longstanding mathematical problems and made significant contributions to the fields of number theory, combinatorics, and theoretical physics.
An Evan Pugh Professor of Mathematics at Pennsylvania State University (University Park, PA), Andrews was elected to the National Academy of Sciences in 2003. He serves on the editorial board for numerous journals, including Discrete Mathematics, Ramanujan Journal, Contemporary Mathematics, Journal of Combinatorial Theory, and Annals of Combinatorics. Andrews was awarded honorary doctorates from the University of Parma (Parma, Italy) in 1998, from the University of Florida (Gainesville, FL) in 2002, and from the University of Waterloo (Waterloo, Ontario, Canada) in 2004. He has also lectured about mathematics in every continent, he says, “except Antarctica.”
Andrews has found an enduring source of research inspiration in a fundamental branch of number theory called partitions, which studies the ways that whole numbers can be split into sums of whole numbers. In his Inaugural Article, published in this issue of PNAS, Andrews explores types of partitions with special restrictions on the summands (1). While developing the topic for his Inaugural Article, Andrews realized that a key lay in a particular function of a class he had studied for his doctoral thesis, called mock theta functions. These functions were first discovered by the Indian mathematician Srinivasa Ramanujan in 1920. “Part of the appeal to me of the Inaugural Article was the fact that Ramanujan's mock theta functions showed up in a setting when I least expected to find them,” Andrews says. In fact, never before has one of the mock theta functions arisen in a partition problem the way it has here. “But I guess I'm used to having surprises,” he adds.
Beginning from a Footnote
Finding unexpected connections in his work is gratifying for a researcher who dreamed as a child of becoming a detective. “When I was in junior high school, I read all of the Sherlock Holmes stories,” Andrews says. “I thought, `Boy, wouldn't it be great if you could spend your life thinking out things and working with your mind?'” As a young man growing up on a farm outside Salem, OR, Andrews thought law was the field most likely to provide him this intellectual freedom, and later he believed he might find satisfaction in engineering. But as an undergraduate in 1956 at Oregon State University (Corvallis, OR), Andrews met an inspiring teacher named Harry Goheen, who was “a bit of a proselytizer for mathematics majors,” Andrews says. Goheen steered him away from engineering, and “I went into mathematics and never looked back,” Andrews says. “I was struck by the beauty and the appeal of mathematics. It just captivated me.”
Ironically, it was outside his mathematics courses where Andrews was first exposed to partition functions. His thenfiancée, Joy, now his wife of 44 years, gave him a fourvolume set of books called The World of Mathematics, which contained Godfrey H. Hardy's A Mathematician's Apology (2). Buried in a footnote was a comment about a surprising mathematical formula that Hardy and Ramanujan had uncovered together in 1916. The elegance of the formula struck Andrews. “I thought it was just stunning,” he says. This fundamental formula in partition theory expresses the number of ways an integer can be broken down into natural number summands. For example, there are three partitions of the number 3 (3, 2+1, and 1+1+1) and five partitions of the number 4 (4, 3+1, 2+2, 2+1+1, and 1+1+1+1). What Hardy and Ramanujan had found was an exact formula for the number of partitions of an integer. “It doesn't seem like you'd need an exact formula,” Andrews explains, “but while there are only five partitions of 4, there are almost 4 trillion partitions of 200.” Even computers could not handle this task as well as an exact formula could, he says, so this result was as useful as it was elegant.
Following Ramanujan's Trail
Andrews graduated from Oregon State University in 1960 with simultaneous bachelor's and master's degrees in mathematics. He spent a year as a Fulbright Scholar in Cambridge, England, and then entered graduate school at the University of Pennsylvania (Philadelphia, PA) in 1961. Although he planned to study a different branch of analytic number theory, prime numbers, he encountered partitions again at the University of Pennsylvania. In a graduate course with esteemed number theorist Hans Rademacher, Andrews saw the ideas of Hardy and Ramanujan's work become fleshed out. “I became so fascinated with what Rademacher was doing that I asked if I could work with him,” he says, and he soon joined Rademacher's group.
Partitions, perhaps deceptively simple at first glance, constitute a rich field of mathematical and academic study, Andrews explains. This is because, in other branches of mathematics, such as number theory, group theory, and algebra, new questions continually arise about special kinds of partitions. Also, partitions are important outside of mathematics: “Of more interest to me is the fact that this process of decomposing a number into summands is such a basic one that it arises in numerous applications, such as in theoretical physics,” Andrews says (3). For example, in statistical mechanics, researchers' investigations hinge on determining the sums of certain fundamental numbers of large systems, called eigenvalues. For a particular case, these eigenvalue sums turn out to be classical partition functions of the type Ramanujan had worked on, Andrews explains.
For Andrews' doctoral research, Rademacher suggested studying another of Ramanujan's pet projects: mock theta functions. Ramanujan began investigating mock theta functions in 1919, for which he left behind no published work. “This topic, which he began right at the end of his life, has been enigmatic all through the 20th century,” Andrews says. The subject of mock theta functions has its roots in classical theta functions, which physicists use in their study of the heat equation. These functions intrigued Ramanujan, who wondered whether they could fit into a wider class of functions, one that was more “subtle,” Andrews says, “but that still retained some of the more important properties of theta functions.”
In 1920, Ramanujan wrote a tantalizing letter to his colleague Hardy, saying he had indeed managed to discover some of these mock theta functions. Yet Ramanujan gave Hardy only a few examples of what he had found. Three months later, Ramanujan passed away. Mathematicians' study of mock theta functions lived on. “No one has really yet found a way of embedding this topic in a wider, fully understood mathematical subject,” Andrews says. “But mock theta functions keep turning up here and there, and interesting questions continue to arise and fascinate people.”
As Rademacher's last student, Andrews wrote his thesis on mock theta functions (4), graduating from the University of Pennsylvania with a doctorate in mathematics after 3 years. He then began to study with number theorists Sarvadaman Chowla and Nathan Fine at Pennsylvania State University in 1964 as an assistant professor. After 3 years, Andrews was granted tenure and promoted to associate professor, and, after 6 years, he reached the rank of full professor. In the interim, he published 27 papers and wrote one book on various aspects of number theory. Andrews turned 32 years old the year he became a full professor—the same age Ramanujan was when he died.
The Lost Notebook
A few years later, in 1975, Andrews visited the University of Wisconsin (Madison, WI) and began work with mathematical analyst Richard Askey, now also a National Academy of Sciences member and professor emeritus at the University of Wisconsin. During his year in Wisconsin, Andrews was invited to speak at a weeklong conference in Strasbourg, France. He took advantage of the trip to Europe to visit Trinity College (Cambridge, U.K.), where he browsed through papers from the estate of the late mathematician George N. Watson, which were housed there.
What Andrews found among the dusty papers grabbed his attention. In one box lay about 100 loose pages filled with, of all things, equations in Ramanujan's handwriting. And there, Andrews realized a few minutes later, were more of the legendary mock theta functions that Ramanujan had hinted at. “It was a gold mine,” Andrews says.
Merely recognizing Ramanujan's handwriting was not the key to Andrews' discovery, his colleague Askey explains. The real detective feat was spotting, in row after row of unlabeled formulas, equations fitting the bill of those enigmatic functions Ramanujan had described. “There were certain identities that George [Andrews] recognized as mock theta functions,” Askey says. “No one else would have spotted them instantly. George has done many things, but this is what will make the history books.”
The collection, hailed by the mathematics community as “the Lost Notebook,” dated from the last year of Ramanujan's life. Containing about 600 equations in all, the notebook eventually revealed to Andrews that Ramanujan had uncovered ideas beyond what he had conveyed to Hardy. In fact, Ramanujan had discovered functions that experts had suggested did not exist. Yet the pages were far from a polished manuscript. Nearly every formula in the lost notebook was stated plainly, Andrews says, without proof. Ramanujan, in his quirky genius, may simply have not seen it necessary to fill in the details of his ideas. Andrews plunged in and started studying this “marvelous collection” (5). He found that only about a fifth of the listed formulas had been independently discovered in the halfcentury since Ramanujan's death.
“So not only am I interviewed, but I'm also sitting in the backseat of the taxicab playing the ghost of G. H. Hardy.”
Working for the past three decades, Andrews, his students, and his colleagues have been able to provide proofs for most of the results. The first volume of the annotated lost notebook, produced by Andrews with Bruce Berndt at the University of Illinois at UrbanaChampaign (Urbana, IL), is currently in the process of being published. “We're hard at work on the second,” Andrews says, with a third volume to follow.
A brief bit of nonmathematical stardom also accompanied Andrews' unearthing of Ramanujan's formulas. In 1987, an episode of the PBS television show “NOVA” related the story of Ramanujan's life. On camera, Andrews recounted his 1975 finding from the Trinity College library. Then, as he later watched the production crew stage a dramatic reenactment of Hardy's taxicab ride to see the ailing Ramanujan in London, Andrews landed another role, too. “All of a sudden, we're ready to go, and they need somebody to sit in the backseat as G. H. Hardy. The only person standing around who doesn't have anything to do is me,” he recalls. “So not only am I interviewed, but I'm also sitting in the backseat of the taxicab playing the ghost of G. H. Hardy.”
A Human Enterprise
Besides mock theta functions, Andrews has tackled a number of other longstanding problems in number theory. In 1990, he put to rest a 200yearold puzzle posed by the Swiss mathematician Leonhard Euler. Euler had described a particular formula that held for the first 10 cases of a series but then failed to be true, which he described as a “misleading induction.” Andrews was able to explain the formula's strange pattern and showed how this answer related to problems in statistical physics (6).
In another case, Andrews solved a problem posed by the physicist Freeman Dyson in 1944. Dyson had coined a new conceptual tool, which he called the “rank,” to explain the behavior of a certain partition function. Dyson had been stymied, however, when he found the rank worked only in some cases but not others. Dyson imagined that there must be an asyetundiscovered, more powerful tool to explain this discrepancy, which he whimsically called the “crank.” In 1987, Andrews and his student Frank Garvan found the crank and fully characterized it (7).
That sort of collaboration is not unusual, Askey says. At times, other researchers—mathematicians, computer scientists, and physicists—will contact Andrews with a problem. “Days later, they get a solution,” Askey says. Andrews enjoys seeing his research contribute to the work of others. “This is certainly partly the obligation of mathematicians,” he says. “But there's also a selfish reason, in that any time you look at your own subject from the point of view of how it can assist others in some seemingly unrelated area, you often find that you learn things about your own subject—new directions and interesting research projects—that had never occurred to you.”
Andrews' interest in collaborating extends to the classroom. He has taught at Pennsylvania State University since 1964, held visiting positions at 15 universities, and supervised 18 doctoral and 16 master's students. He has instructed students in mathematics at every level, from calculus to advanced topics for graduate students. In 1981, he was appointed as the Evan Pugh Professor of Mathematics, chosen for excellence in both research and teaching. “He's a marvelous teacher,” Askey says.
One of Andrews' auxiliary passions is the quality of mathematics education today, from elementary school onward. “Fundamentally, my feeling is that many, if not all, of the reforms undertaken in primary and secondary education have only made problems worse,” he says. He agrees with the writer Ernst F. Schumacher in that the challenge education faces is a “divergent” problem, to borrow a term from mathematics. That is, how can a teacher reconcile two opposite needs in a classroom: authority and discipline to effectively impart knowledge versus the freedom to be creative? The reconciliation must come from the more refined skills, he says, such as understanding and compassion (8). “I believe that, while mathematics may indeed be the queen of sciences, teaching mathematics is a very human enterprise,” Andrews says. “It is an art and not a science in any way, and it brings forth all of the virtues and flaws of humanity.”
Discoveries Ahead
Presently, Andrews is on sabbatical, dividing his time working at the University of Linz (Linz, Austria) and the University of Florida, where he is to give a series of lectures on the work of Ramanujan (9). Andrews is also collaborating with Peter Paule and Axel Reise in Austria on a software project called omega, which aids research in the theory of partitions (10). omega is able to produce answers to otherwise unmanageable problems fairly quickly, he explains.
The research presented in Andrews' Inaugural Article leaves many open questions and is just the start of a new project, he says. Indeed, his whole career leaves plenty of opportunity for the unexpected to cross his path. “It seems to me that there's this grand mathematical world out there, and I am wandering through it and discovering fascinating phenomena that often totally surprise me,” he says. “I do not think of mathematics as invented, but rather discovered.”
Footnotes

This is a Biography of a recently elected member of the National Academy of Sciences to accompany the member's Inaugural Article on page 4666.
 Copyright © 2005, The National Academy of Sciences
References

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Andrews, G. E. (2005) Proc. Natl. Acad. Sci. USA 102 , 46664671. pmid:15716357

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Hardy, G. H. (1940) A Mathematician's Apology (Cambridge Univ. Press, London).

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Andrews, G. E., Baxter, R. J. & Forrester, P. J. (1984) J. Stat. Phys. 35 , 193266.

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Andrews, G. E. (1966) Am. J. Math. 88 , 454490.

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Andrews, G. E. (1979) Am. Math. Mon. 86 , 89108.

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Andrews, G. E. (1990) J. Am. Math. Soc. 3 , 653669.

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Andrews, G. E. & Garvan, F. (1988) Bull. Am. Math. Soc. 18 , 167171.

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Andrews, G. E. (2001) Am. Math. Mon. 108 , 281285.
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