Bhargava
strikes balance among many interests
http://www.princeton.edu/pr/pwb/03/1208/1b.shtml
By Steven Schultz
Princeton
NJ -- Mathematician Manjul Bhargava has unraveled 200- year-old
problems and catapulted to the top of his field, but right
now he is not thinking about equations or prime numbers.
He is sitting cross-legged 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 wire-rimmed
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 math
Bhargava 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 centuries-old 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 long-term 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
applications
The 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
real-world 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 overview
His goals as a teacher began to take shape when he was a
graduate student and served on a committee charged with
re-evaluating 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
easy-going 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."