Raoul Bott at Harvard*,† by Shing-Tung Yau‡ and Steve Nadis§ The hiring of Raoul Bott also paid off immensely when World War II came to a sudden and unexpected for Harvard, although it is fair to say that as a young end. man, Bott did not exhibit great mathematical flair, nor Bott then enrolled in a one-year master’s program did he show much academic promise in general. Born in engineering at McGill, but after completing that he in Budapest in 1923—and raised mainly in Slovakia was still unsure of his future course. So he met with (until his family immigrated to Canada in 1938)—Bott the dean of McGill’s medical school, as he contem- was, at best, a mediocre student throughout child- plated a career shift. In response to questions posed hood. In five years of schooling in Bratislava, Slovakia, by the dean, Bott confessed to disliking the dissection he did not earn a single A, except in singing and Ger- of animals and hating chemistry even more, showing man. In mathematics, he typically got Cs and the oc- little enthusiasm for the standard subjects taken up casional B, which should make him a hero among late in medical school. Finally, the dean asked him: “Is it bloomers. that maybe you want to do good for humanity…? Be- 3 As a youth of about twelve to fourteen, Bott and cause they make the worst doctors.” a friend had fun playing around with electricity—cre- That ended Bott’s thoughts of a medical career. “I ating sparks, wiring together fuse boxes, transform- thanked him,” Bott said. “And as I walked out of his ers, and vacuum tubes, and, in the process, figuring door I knew that I would start afresh and with God’s 4 out how various gadgets work. This experimentation grace try and become a mathematician.” eventually served him well. A mathematician, Bott Initially, he wanted to pursue mathematics at later explained, is “someone who likes to get to the McGill, until he was told that his background was so root of things.”1 thin he would have to get a bachelor’s degree first—a process that could take three years. He turned in- Although Bott frequently told his Harvard stu- stead to the Carnegie Institute of Technology (since dents that he never would have made it into the renamed Carnegie Mellon University), interested in school as an undergraduate, he somehow managed the master’s program in applied mathematics. But the to get into McGill University, where he majored in course requirements were so extensive that it would electrical engineering:2 Upon graduating in 1945, he have taken him three years to get a master’s degree. joined the Canadian army but left after four months Carnegie’s math chair, John Lighton Synge, suggested * Excerpted from the book A History in Sum: 150 Years their new doctoral program, which had hardly any re- of Mathematics at Harvard (1825–1975), by Steve Nadis quirements at all. Bott liked the idea and was assigned and Shing-Tung Yau, published by Harvard University Press, Richard J. Duffin as his adviser. 2013. Copyright © 2013 by the President and Fellows of Har- Bott and Duffin took on, and eventually solved, vard College. Used by permission. All rights reserved. † (The title used here, “Raoul Bott at Harvard”, does not ap- what was then one of the most challenging problems pear in the original text, and the original bibliographic cita- in electrical network theory. The resultant Bott-Duffin tions have been re-numbered.) theorem not only was of great theoretical significance ‡ Department of Mathematics, Harvard University but also had important practical applications in the E-mail: [email protected] electronics industry. The paper that Bott and Duffin § Steve Nadis is a freelance science writer living in Cam- bridge, Massachusetts coauthored had important consequences for Bott’s E-mail: [email protected] career, as well, because the work impressed Hermann JULY 2014 NOTICES OF THE ICCM 87 Weyl, who arranged for Bott to spend the 1949–50 “Generally, the critical points of a function are iso- academic year at the Institute for Advanced Study in lated,” Bott wrote, but he realized these points could Princeton:5 (Weyl, as you may recall, came up earlier come in “bigger aggregates” and could even be the in this chapter, having secured a position at the In- special kinds of spaces we call manifolds:9 stitute for Advanced Study for Richard Brauer; Weyl One way to picture this is to take the upright later did the same for Richard’s brother, Alfred.) doughnut from the previous example and topple it “The general plan of my appointment [at the in- over, so that it is now lying on its side, flat on a table- stitute], as I understood it,” Bott wrote, “was that I top. The maximum of this newly configured object is was to write a book on network theory at the Insti- no longer a point—it is a circle. The minimum, simi- tute.” On his first day at work in Princeton, Bott met larly, is a circle too. One could determine the topology with Marston Morse, who was in charge of the tem- of the space—and correctly identify it as a torus—by porary members that year. “[Morse] immediately dis- knowing that the critical manifolds are two circles, missed my fears of having to write a book. It was a aligned one above the other. matter of course to him that at the Institute a young Bott thus provided a generalization of classi- man should only do what he wanted to do; that was cal Morse theory, often called Morse-Bott theory, in the place where a young man should find himself and which critical manifolds replaced the critical points the last place in the world for performing a chore… of the original theory. The critical manifolds of this I remember leaving this interview with a light heart, theory could be individual points, which is the zero- newly liberated, and buoyed by the energy and opti- dimensional special case. Or they could be one- mism I had just encountered.”6 dimensional manifolds, like the circles in the tipped doughnut. They could be higher-dimensional objects, Bott immediately went to his office and started too—manifolds of any finite dimension, in fact. working on the four-color problem, thinking that the Bott used this generalized version of Morse the- trick he had used to solve the network problem—a ory to compute the homotopy groups of a manifold, function that he considered his “secret weapon,” and from there he proved the periodicity theorem. might crack this problem as well. Morse dropped by That, admittedly, is a bit of a mouthful, so we will for a chat a few weeks later. “When he heard what try to break down that statement, explaining in sim- I was doing, he didn’t really object,” Bott wrote. “In- plified terms what he did. stead, he first spoke of the great interest in the ques- Whereas Morse was primarily interested in us- tion, but then started to talk about the many good ing topology to solve problems in analysis—to solve men he had seen start to work on it, never to reap- differential equations, in other words—Bott turned pear again. After he left, I threw all my computations that around, using Morse theory to solve problems in the waste basket and never thought about the ques- in topology. And one of the main problems in topol- tion again!”7 ogy—as in other areas of mathematics and through- What he did think about—perhaps owing to out science, in general—is the classification problem. the presence of Morse, Norman Steenrod, and oth- “Just as scientists want to classify plants and ani- ers—was topology. Surrounded by giants in the field, mals to understand how biology works and how life Bott studied the subject deeply, though he appeared is or organized, mathematicians also strive to find to be in no great rush to publish anything. In fact, some order among mathematical objects,” says Tufts he said, “I didn’t write a single paper in my first year University mathematician Loring Tu, a former Har- there. So I was very delighted when Marston Morse vard graduate student who coauthored a book on al- called me up at the end of that year and said, ‘Do gebraic topology with Bott.10 Group theorists, there- you want to stay another year?’ And I said, ‘Of course, fore, are interested in classifying groups, such as the yes!’ He said, ‘Is your salary enough?’ (It was $300 a finite simple groups discussed earlier in this chap- month.) I said, ‘Certainly!’ because I was so delighted ter. Topologists, similarly, want to be able to look to be able to stay another year. My wife took a dimmer at various spaces—seeing which ones are equivalent, 8 view, but we managed.” which ones are different—and then sort them into In 1951, Bott joined the faculty of the University their proper bins. of Michigan, where he continued to focus on topol- Mathematicians define invariants—fixed, intrin- ogy, paying particular attention to Morse’s theory of sic features of a space—in order to distinguish among critical points. A standard picture from Morse the- different topological spaces. If two spaces are “home- ory, as discussed in Chapter 4, involves a doughnut omorphic”—meaning that one can be deformed into (or “torus”) standing upright. This surface has four the other by stretching, bending, or squishing but not “critical points”—a maximum on top of the dough- cutting—they must have the same topological invari- nut, a minimum on the bottom, and two saddle points ants.