Mathematicians have found a framework
for the celebrated Rogers-Ramanujan
identities and their arithmetic
properties, solving another long-
standing mystery stemming from the work of
Indian math genius Srinivasa Ramanujan.
The findings, by mathematicians at Emory
University and the University of Queensland,
yield a treasure trove of algebraic numbers and
formulas to access them.
"Algebraic numbers are among the first
numbers you encounter in mathematics," says
Ken Ono, a number theorist at Emory "And yet,
it's surprisingly difficult to find functions that
return them as values in a uniform and
systematic way."
Ono is the co-author of the new findings,
along with S. Ole Warnaar of the University of
Queensland and Michael Griffin, an Emory
graduate student.
Ono announced the findings in April as a
plenary speaker at the Applications of
Automorphic Forms in Number Theory and
Combinatorics conference at Louisiana State
University. He will also present them as a
plenary speaker at the 2015 Joint Mathematics
Meetings, the largest mathematics meeting in
the world, set for January in San Antonio.
Warnaar, Griffin and others will give additional
talks on the findings during an invited special
session to accompany Ono's plenary address.
The most famous algebraic number of all is the
golden ratio, also known by the Greek letter
phi.
Many great works of architecture and art,
such as the Parthenon, are said to embody the
pleasing proportions of the golden ratio, which
is also seen in beautiful forms in nature.
Mathematicians, artists and scientists, from
ancient times to today have pondered the
qualities of phi, which is approximately equal
to 1.618, although its digits just keep on going,
with no apparent pattern.
"People studied the golden ratio before there
was a real theory of algebra," Ono says. "It
was a kind of prototype for algebraic
numbers."
Although no other algebraic units are as
famous as the golden ratio, they are of central
importance to algebra. "A fundamental
problem in mathematics is to find functions
whose values are always algebraic numbers,"
Ono says. "The famous Swiss mathematician
Leonhard Euler made some progress on this
problem in the 18th century. His theory of
continued fractions, where one successively
divides numbers in a systematic way, produces
some very special algebraic numbers like the
golden ratio. But his theory cannot produce
algebraic numbers which go beyond the stuff
of the quadratic formula that one encounters
in high school algebra."
Ramanujan, however, could produce such
numbers, and he made it look easy.
"Ramanujan has a very special, almost mythic,
status in mathematics," says Edward Frenkel, a
mathematician at the University of California,
Berkeley. "He had a sort of Midas touch that
seemed to magically turn everything into gold."
And the Rogers-Ramanujan identities are
considered among Ramanujan's greatest
legacies, adds Frenkel, a leading expert on the
identities.
"They are two of the most remarkable and
important results in the theory of q-series, or
special functions," says Warnaar, who began
studying the Rogers-Ramanujan identities
shortly after he encountered them while
working on his PhD in statistical mechanics
about 20 years ago.
Although originally discovered by L. J. Rogers
in 1894, the identities became famous through
the work of Ramanujan, who was largely self-
taught and worked instinctively.
In 1913, Ramanujan sent a letter from his
native India to the British mathematician G. H.
Hardy that included the two identities that
Rogers discovered and a third formula that
showed these identities are essentially modular
functions and their quotient has the special
property that its singular values are algebraic
integral units. That result came to be known as
the Rogers-Ramanujan continued fraction.
Hardy was astonished when he saw the
formulas. "I had never seen anything in the
least like this before," Hardy wrote. "A single
look at them is enough to show they could
only be written down by a mathematician of
the highest class. They must be true because
no one would have the imagination to invent
them."
"Ramanujan seemed to produce this result out
of thin air," Ono says.
Ramanujan died in 1920 before he could
explain how he conjured up the formulas.
"They have been cited hundreds of times by
mathematicians," Ono says. "They are used in
statistical mathematics, conformal field theory
and number theory. And yet no one knew
whether Ramanujan just stumbled onto the
power of these two identities or whether they
were fragments of a larger theory."
For nearly a century, many great
mathematicians have worked on solving the
mystery of where Ramanujan's formulas came
from and why they should be true.
Ono uses the analogy of going for a walk in a
creek bed and discovering a piece of gold. Had
Ramanujan accidentally found a random
nugget? Or was he drawn to that area because
he knew of a rich seam of gold nearby?
Warnaar was among those who pondered these
questions. "Just like digging for gold, in
mathematics it's not always obvious where to
look for a solution," he says. "It takes time
and effort, with no guarantee of success, but it
helps if you develop a lot of intuition about
where to look."
Finally, after 15 years of focusing almost
entirely on the Rogers-Ramanujan identities,
Warnaar found a way to embed them into a
much larger class of similar identities using
something known as representation theory.
"Ole found the mother lode of identities," Ono
says.
When Ono saw Warnaar's work posted last
November on arXiv.org, a mathematics-physics
archive, his eyes lit up.
"It just clicked," Ono recalls. "Ole found this
huge vein of gold, and we then figured out a
way to mine the gold. We went to work and
showed how to come full circle and make use
of the formulas. Now we can extract infinitely
many functions whose values are these
beautiful algebraic numbers."
"Historically, the Rogers-Ramanujan identities
have tantalized mathematicians," says George
Andrews, a mathematician at Penn State and
another top authority on the identities.
"They
have played an almost magical role in many
areas of math, statistical mechanics and
physics."
The collaboration of Warnaar, Ono and Griffin
"has given us a big picture of the general
setting for these identities, and deepened our
theoretical understanding for many of the
breakthroughs in this area of mathematics
during the past 100 years,"
Andrews says.
"They've given us a whole new set of tools to
be able to attack new problems."
"It's incredibly exciting to solve any problem
related to Ramanujan, he's such an important
figure in mathematics," Warnaar says. "Now
we can move on to more questions that we
don't understand. Math is limitless, and that's
fantastic."
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