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Thanks to a group of sharp-minded, hardworking students, Carl M. Bender, Ph.D., has purged many mistakes from his book-in-progress. "I'm handing it out chapter by chapter as I write it," said Bender, professor of physics in Arts and Sciences. "It makes the course more interesting, the book better and the students eligible for rewards. Last night I received e-mail from someone who is getting two extra points for finding an error."
The textbook in the making, "Partial Differential Equations for Scientists and Engineers," presents advanced mathematics to aspiring physicists, engineers, and mathematicians. "The publication date is June '98, when I finish the course," Bender said. "It'll be the big bang."
It's vintage Bender. Puns and cornball jokes. An office annotated with cartoons and creative signage. A piece of paper taped to one wall reads "dry paint." The levity lightens up the highly complex, head-spinning, equation-crunching world of modern physics. "This is fancy stuff," said Bender, "a field where the task of writing down a single meaningful equation is intensely difficult."
A theoretical and particle physicist, Bender works primarily in quantum mechanics, the physics of very small, submicroscopic or atomic particles, and in particle physics and quantum field theory. Quantum mechanics underlies nearly all modern science and technology; it governs the essential components of computers and TVs and serves as a basis for chemistry.
Particle physics is devoted to identification and description of matter and its properties, such as mass and charge, and what holds it together.
"There are two kinds of theoreticians," Bender explained. "The phenomenological theoretician talks to scientists who design the experiments. The theoretical theoretician stands back from the theory to think about its nature and formulation."
Bender serves in the latter capacity, exploring the boundary between physics and his passion, mathematics. "I use science as a way of generating the problems," he said.
Bender's work has generated a great deal more. "Carl is one of our leading lights in the department," said Clifford M. Will, Ph.D., professor and chair of physics. "Not only is he a world-class physicist, but also one of the best teachers on campus. He excites and inspires students. He serves on many University committees and has made Washington University known throughout America for its high and consistent performance in the Putnam competition."
A Putnam coach for 20 years, Bender and mathematics Professor Richard Rochberg, Ph.D., helped place the University among the top three schools of the 400 U.S. and Canadian institutions participating in the prestigious mathematical examination for undergraduates. In the past two decades, the University has placed first and second four times each.
Also the ombudsman for the College of Arts and Sciences, Bender serves on a half dozen University and departmental committees.
Working in England and Israel
He has earned many distinctions, among them a Fulbright fellowship to lecture and conduct research at Imperial College in London from summer 1995 through summer 1996. He received additional support from the Particle Physics and Astronomy Research Council (PPARC), the British equivalent of the National Science Foundation (NSF).
"It was an amazing and wonderful time," recalled Bender, who had traveled to Imperial College in 1974 as a visiting fellow and again in 1986 as a visiting professor. "It gave me the chance to talk about my work and to collaborate with others. What great fun to sit all day and think about problems, work straight through lunch, not even put down the pencil.
"I talked with many students and must have solved 10 or 12 different problems, two or three times my normal annual production," he added.
During his trip abroad, Bender also taught and conducted research at Technion-Israel Institute of Technology in Haifa, Israel, on a Lady Davis Fellowship, a coveted award given to only a few visiting scholars each year for study in Israel. Another highlight was a month spent at Cambridge University's Newton Institute, which hosted a six-month conference on hyperasymptotics, a field of mathematical physics that Bender helped create through his early work on perturbation theory.
"Perturbation theory involves an organized mathematical set of procedures for solving very difficult problems that bear similarity to solvable ones," Bender explained. "For example, one problem we can solve is the shape of Earth's orbit as the planet moves around the sun, but only if there are no other planets. The pull of gravity from the other planets disturbs the calculations. These problems can be solved approximately but never exactly."
Bender's first book, "Advanced Mathematical Methods for Scientists and Engineers" (1978, McGraw-Hill), co-authored with Princeton Professor S.A. Orszag, has become the classic work on perturbation theory and is used by universities stretching from Harvard to California Institute of Technology. Bender credits his wife, Jessie, "a highly talented editor," with contributing to the book's success.
"I grew up quantitative," Bender said, moving along a banquet of published papers that runs nearly the length of his Compton Hall office. He points out an article co-authored with Jade Vinson, now attending Princeton on a National Defense Science and Engineering Graduate fellowship.
Vinson worked with Bender -- "a mentor I still go to for advice," Vinson noted -- beginning the summer after his freshman year. "He's energetic, patient, and open-minded," Vinson said of his former professor. "He is known for his deep involvement with undergraduates."
Another paper presents a methodology for studying a specific geyser. Bender co-authored this article with his son, Daniel, a third-year graduate student studying the history of the American labor movement at New York University. (His older son, Michael, will receive a Ph.D. in computer science from Harvard next spring, when he plans also to marry.)
"I always thought I would be a chemist. In high school, I had a lab in my basement, where I was always discovering things," Bender related. "My father was a physics teacher who moonlit as a TV repairman. I was the radio component of his business."
As a freshman at Cornell University, Bender immediately enrolled in an upper-level chemistry course and earned the highest grade in the class. He excelled in every course he took. Thrilled by the discovery of a water-soluble glass he developed over the subsequent summer, Bender knocked on a professor's door to share the find.
"I talked about the applications the glass might have in medical equipment," Bender recalled. "The professor looked up at me and said, 'Why don't you check it out at the library?' I lost it then. They were too absorbed with their graduate students."
Yet for every disappointing professor Bender encountered, he found a prize. At Cornell, it was Ken Greisen, who advised Bender, by now disenchanted with experimental chemistry, to try physics. "I was just blowing with the wind then, taking courses randomly, loving the science and math," Bender said. "During graduation week, I got a call from the math department asking if I was a math major. I didn't know."
Bender graduated summa cum laude with a bachelor's degree in physics and entered Harvard University "as a serious theoretical physicist." Harvard professor Julian Schwinger had won the 1965 Nobel Prize for Physics (with Richard Feynman and Sin-Itiro Tomonaga), and Bender was hoping to work with him.
Disappointment closed in on the prospect, however, as Schwinger never seemed accessible. After several weeks, Bender made it into his office, but the encounter was all too brief and the result a letdown. "I asked the great man my question," said Bender. "He answered, 'I don't know.'"
Enter T.T. Wu, applied mathematician and physicist, who became Bender's thesis adviser, collaborator and co-author on several papers. "Wu taught me how to think about a problem, how to attack the unknown," Bender said. "It was an approach more than anything. He showed me how to shake hands with a problem."
Following a year of postdoctoral work at Princeton, Bender took a position in the applied mathematics department at the Massachusetts Institute of Technology, where he stayed for seven years. But when Washington University offered him a position, he left MIT. "My productivity increased here," he said, "because I spent less time worrying about the noise around me. I found room in my mind to work on larger and more long-range problems. The classes were smaller and the students brighter. My whole life changed."
As a result, the lives of others have changed, too. Junior Michael Lamar recalled one of Bender's lectures presented when he visited the University as a high school senior. "He [Bender] captivated the audience and led to my enrollment here," said Lamar, a student in Bender's advanced mathematics course. "He's an excellent teacher who livens the moment."
He is a fellow of the American Physical Society, serves on professional journals' editorial boards and has been a consultant at Los Alamos National Laboratory in New Mexico since 1979. ("I don't make bombs," he quickly interjected.)
Bender's research has been funded annually by the Department of Energy since he joined the University two decades ago. He has built a vibrant area in theoretical physics that now includes five professors, a research professor and a postdoctoral researcher. Two additional scientists who have received funding to work specifically with Bender's group will arrive next year.
Bender took from his shelf a yellowed bound copy of his thesis on the analytic properties of the anharmonic oscillator. "You'll like appendix H," he said, handing over the volume.
What did the dissertation readers make of this?
"No one caught it, which made me wonder if they'd read it [the thesis]," said Bender. "I showed them afterward."
"They laughed, too."
-- Cynthia Georges
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