214 Notices of the AMs VoluMe 56, NuMber 2
Before quantum mechanics was invented, classical
physics was always nonlinear, and linear models
were only approximately valid. After quantum
mechanics, nature itself suddenly became linear.
This had profound consequences for mathemat-
ics. During the nineteenth century Sophus Lie
developed his elaborate theory of continuous
groups, intended to clarify the behavior of classical
dynamical systems. Lie groups were then of little
interest either to mathematicians or to physicists.
The nonlinear theory of Lie groups was too compli-
cated for the mathematicians and too obscure for
the physicists. Lie died a disappointed man. And
then, fifty years later, it turned out that nature was
precisely linear, and the theory of linear represen-
tations of Lie algebras was the natural language of
particle physics. Lie groups and Lie algebras were
reborn as one of the central themes of twentieth
century mathematics.
A third joke of nature is the existence of quasi-
crystals. In the nineteenth century the study of
crystals led to a complete enumeration of possible
discrete symmetry groups in Euclidean space.
Theorems were proved, establishing the fact that
in three-dimensional space discrete symmetry
groups could contain only rotations of order three,
four, or six. Then in 1984 quasi-crystals were dis-
covered, real solid objects growing out of liquid
metal alloys, showing the symmetry of the icosa-
hedral group, which includes five-fold rotations.
Meanwhile, the mathematician Roger Penrose
discovered the Penrose tilings of the plane. These
are arrangements of parallelograms that cover a
plane with pentagonal long-range order. The alloy
quasi-crystals are three-dimensional analogs of
the two-dimensional Penrose tilings. After these
discoveries, mathematicians had to enlarge the
theory of crystallographic groups to include quasi-
crystals. That is a major program of research which
is still in progress.
A fourth joke of nature is a similarity in be-
havior between quasi-crystals and the zeros of
the Riemann Zeta function. The zeros of the zeta-
function are exciting to mathematicians because
they are found to lie on a straight line and nobody
understands why. The statement that with trivial
exceptions they all lie on a straight line is the
famous Riemann Hypothesis. To prove the Rie-
mann Hypothesis has been the dream of young
mathematicians for more than a hundred years.
I am now making the outrageous suggestion that
we might use quasi-crystals to prove the Riemann
Hypothesis. Those of you who are mathematicians
may consider the suggestion frivolous. Those who
are not mathematicians may consider it uninterest-
ing. Nevertheless I am putting it forward for your
serious consideration. When the physicist Leo
Szilard was young, he became dissatisfied with the
ten commandments of Moses and wrote a new set
of ten commandments to replace them. Szilard’s
second commandment says: “Let your acts be di-
rected towards a worthy goal, but do not ask if they
can reach it: they are to be models and examples,
not means to an end.” Szilard practiced what he
preached. He was the first physicist to imagine
nuclear weapons and the first to campaign ac-
tively against their use. His second commandment
certainly applies here. The proof of the Riemann
Hypothesis is a worthy goal, and it is not for us to
ask whether we can reach it. I will give you some
hints describing how it might be achieved. Here I
will be giving voice to the mathematician that I was
fifty years ago before I became a physicist. I will
talk first about the Riemann Hypothesis and then
about quasi-crystals.
There were until recently two supreme unsolved
problems in the world of pure mathematics, the
proof of Fermat’s Last Theorem and the proof of
the Riemann Hypothesis. Twelve years ago, my
Princeton colleague Andrew Wiles polished off
Fermat’s Last Theorem, and only the Riemann Hy-
pothesis remains. Wiles’ proof of the Fermat Theo-
rem was not just a technical stunt. It required the
discovery and exploration of a new field of math-
ematical ideas, far wider and more consequential
than the Fermat Theorem itself. It is likely that
any proof of the Riemann Hypothesis will likewise
lead to a deeper understanding of many diverse
areas of mathematics and perhaps of physics too.
Riemann’s zeta-function, and other zeta-func-
tions similar to it, appear ubiquitously in number
theory, in the theory of dynamical systems, in
geometry, in function theory, and in physics. The
zeta-function stands at a junction where paths lead
in many directions. A proof of the hypothesis will
illuminate all the connections. Like every serious
student of pure mathematics, when I was young I
had dreams of proving the Riemann Hypothesis.
I had some vague ideas that I thought might lead
to a proof. In recent years, after the discovery of
quasi-crystals, my ideas became a little less vague.
I offer them here for the consideration of any
young mathematician who has ambitions to win
a Fields Medal.
Quasi-crystals can exist in spaces of one, two,
or three dimensions. From the point of view of
physics, the three-dimensional quasi-crystals are
the most interesting, since they inhabit our three-
dimensional world and can be studied experi-
mentally. From the point of view of a mathemati-
cian, one-dimensional quasi-crystals are much
more interesting than two-dimensional or three-
dimensional quasi-crystals because they exist in
far greater variety. The mathematical definition
of a quasi-crystal is as follows. A quasi-crystal
is a distribution of discrete point masses whose
Fourier transform is a distribution of discrete
point frequencies. Or to say it more briefly, a
quasi-crystal is a pure point distribution that has
a pure point spectrum. This definition includes