Bhargava strikes balance among many interestsBy Steven Schultz Princeton NJ  Mathematician Manjul Bhargava has unraveled 200 yearold problems and catapulted to the top of his field, but right now he is not thinking about equations or prime numbers. He is sitting crosslegged before a tabla, a pair of small Indian drums, and letting his fingers fly. A rhythm rolls forth  enveloping, evolving, precise and melodic. His boyish face glows with delight. He has come, in traditional Indian dress, to the top floor of Fine Hall to have his picture taken as he plays. For a few minutes, his office one floor below seems far away, but the spark behind his wirerimmed glasses is the same one that flashes other times when he describes his math and the beauty of ideas that fit together like parts of a musical score. Bhargava, 28, joined the Princeton faculty this fall as a full professor of mathematics, one of the youngest ever to receive that rank, and brings passion to many pursuits. In addition to his math and music, Bhargava is committed to teaching and has ambitious plans for developing new introductory math courses that broaden the subject's appeal. He also has written articles to popularize math for a general audience and published research in the field of linguistics. "He is an amazingly talented young guy," said Professor of Mathematics Peter Sarnak, who has worked with him since Bhargava was a graduate student at Princeton from 1996 to 2001. Bhargava's varied interests build on one another, but not always directly. Classical Indian music is very mathematical, but consciously thinking of the math would interfere with the improvisation and emotion of the playing, Bhargava said. "But somehow the connection is there. I often use music as a break, and many times I come back to the math later and things have cleared up," he said. Even when he is working on his research, mathematics is not necessarily on his mind. "I am always working on lots of things at once," he said. "I like to move between different areas. "Often I am not thinking about any particular problem at all," he added. "I am just thinking, wherever it leads. Often the problem being solved emerges only later." Music and mathBhargava was born in Canada and grew up on Long Island, the son of immigrants from the Jaipur region of India. He remains close to his Indian roots. One of his greatest influences, he said, was his grandfather, a prominent linguist and scholar of ancient Indian history who gave Bhargava training in Sanskrit. It was his mother, though, a professor of mathematics at Hofstra University, who exposed him to mathematics. Bhargava remembers looking for a formula to account for the spacing of prime numbers (a problem that continues to be one of the central quandaries of mathematics) when he was in second grade. His mother, who plays the tabla and other Indian instruments, also introduced him to classical Indian music. She showed him the most basic note, called "na," on the tabla, a surprisingly difficult snap of the forefinger on the edge of one drum. "I remember when I was three, I heard my mom and tried to copy her. That's one of the things that drew me into the tabla: I had to learn to make that sound." By the time he was 12, Bhargava showed considerable talent and spent time in India studying the tabla. He continues to study with two of the world's most accomplished tabla masters, Pandit Prem Prakash Sharma and Ustad Zakir Hussain. He considered music as a career but always maintained his love of mathematics. He majored in math at Harvard University, where he quickly began producing substantial research and published several papers. He won the American Mathematical Society's Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student and graduated as salutatorian. In graduate school at Princeton, Bhargava focused on the area of mathematics called number theory, working as the advisee of Andrew Wiles. Wiles suggested that Bhargava tackle problems that grew out of work done by Johann Gauss, who is one of history's greatest mathematicians and founded the field of algebraic number theory in 1801. Bhargava took Gauss' work much further than he or anyone else in the department thought possible. Gauss had found a method for combining two quadratic equations (equations with a form like x2 + 2xy + 5y2 = 0) in a way that was very different from normal addition and revealed a lot of information about number systems. Mathematicians have been studying Gauss' "composition law," as it was called, ever since. Although progress has been made on understanding many aspects of Gauss' work, one line of questioning "came to a stop with Gauss" and had not yielded progress in 200 years, said Wiles. Bhargava not only broke new ground in that area, he discovered 13 more composition laws and developed a coherent mathematical framework to explain them. "It took everyone completely by surprise," said Wiles. Bhargava's approach was also notable because it was elegant and "very classical," said Wiles, who also has solved a centuriesold problem, Fermat's Last Theorem. "He did it in a way that Gauss himself could have understood and appreciated." "It was stunning," said Sarnak. Bhargava's work earned him a prestigious longterm fellowship with the Clay Mathematics Institute of Cambridge, Mass. The appointment funds research, travel and salary expenses for five years. The magazine Popular Science also recognized his achievements, choosing Bhargava for its "Brilliant 10" list in 2002. Practical applicationsThe fundamental nature of Bhargava's work already has caused it to have a major impact on the field of number theory and assures that it will continue to do so for many years, his colleagues said. It also could have practical applications. Number theory, which involves only integers, such as 1, 2, 3, and not fractions or decimals, used to be considered an area of mathematics so pure that it would never find uses in realworld problems. But that changed with the computer age in which digital information, like the integers themselves, comes in discrete packets. Number theory, for example, now plays a central role in cryptography, which is critical for tasks such as keeping information secure on the Internet. Before graduate school, Bhargava worked at the U.S. government's Center for Communications Research in Princeton, where he applied number theory to questions of cryptography. That work is closely related to his current research and continues to be one of the many ideas on his mind. During his graduate work, Bhargava solved several other significant problems, including one he worked out in part of his thesis, which Sarnak noted would easily have earned him a Ph.D. on its own. He also renewed an interest kindled by his grandfather and he began publishing research on linguistics, a subject he continues to pursue. After graduating, he spent a year at the Institute for Advanced Study before taking visiting lectureships at Harvard and Princeton, both of which were recruiting him to join their faculties. A major factor in tipping Bhargava's decision toward Princeton was the opportunity to have a significant impact on the teaching of mathematics. He fell in love with teaching as an undergraduate at Harvard, where he lectured and led precepts during his sophomore, junior and senior years. He won Harvard's Derek Bok Award for Excellence in Teaching each of those three years. Bhargava also has worked to enliven mathematics for audiences beyond university students by writing articles in nontechnical magazines and journals. This writing earned him the 2003 Merten Hasse Prize for Exposition from the American Mathematical Society. A greater overviewHis goals as a teacher began to take shape when he was a graduate student and served on a committee charged with reevaluating the undergraduate math curriculum at Princeton. The committee found that the department was losing talented students to other majors in part because its principal introductory course was too focused on one area of mathematics. Princeton students do not traditionally see other mathematical specialties or applied aspects of mathematics until their junior and senior years. "It was clear it was a great course," said Bhargava. "Many loved it and went on to become math majors. But there were many who didn't enjoy the subject content, because it did not address their particular interests." Bhargava, who is teaching a graduate research seminar this semester, is developing an introductory course titled "An Introduction to Mathematical Thinking," which will provide a greater overview of the field than currently is available. He hopes the course will appeal to prospective math majors as well as students in other departments. Engineering Dean Maria Klawe, who helped recruit Bhargava to Princeton, said she plans to collaborate with him in developing new math courses for engineering students, who also would benefit from seeing interesting applications during their introductory courses. "There are many ways to appreciate mathematics, and we need to provide alternate paths to students," she said. "That is something that Manjul really understands." Nicholas Katz, chair of Princeton's math department, also said Bhargava's combination of interests will be very valuable to the department. "It's not always easy to find  in young people who are having very active and exciting scientific lives  this interest in and commitment to reaching out to students and getting them interested in math. So when you find someone who combines great scholarship with great teaching, it's just wonderful." Bhargava's return to Princeton also was welcomed in other areas of the University. Last spring he collaborated with Professor of Computer Science Perry Cook and Assistant Professor of Music Dan Trueman on a performance of digital music at an international music conference that took place across the Internet with some musicians in Montreal and others in Princeton. Trueman said he hopes to collaborate with Bhargava to bring more cultural diversity to the University's music curriculum. Those who have worked with him said Bhargava's enthusiasm and easygoing manner make collaborating with him a joy. "I met him at our rehearsal and thought he was a professional tabla player who was hired in to play the gig," said Trueman. "I found out later he was a famous mathematician. It's a wonderful thing to have him here on the faculty." 
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