This is an excerpt from F. Dyson, “Birds and Frogs”, Notices of the AMS (February 2009) 212–233 [a written version of his AMS Einstein Lecture, which was to have been given in October 2008 but which had to be cancelled]
 

“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 Hypothesis remains. Wiles’ proof of the Fermat Theorem was not just a technical stunt. It required the discovery and exploration of a new field of mathematical 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-functions 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 threedimensional world and can be studied experimentally. From the point of view of a mathematician, one-dimensional quasi-crystals are much more interesting than two-dimensional or threedimensional 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 as a special case the ordinary crystals, which are periodic distributions with periodic spectra.

Excluding the ordinary crystals, quasi-crystals in three dimensions come in very limited variety, all of them associated with the icosahedral group. The two-dimensional quasicrystals are more numerous, roughly one distinct type associated with each regular polygon in a plane. The twodimensional quasi-crystal with pentagonal symmetry is the famous Penrose tiling of the plane. Finally, the onedimensional quasi-crystals have a far richer structure since they are not tied to any rotational symmetries. So far as I know, no complete enumeration of one-dimensional quasi-crystals exists. It is known that a unique quasi-crystal exists corresponding to every Pisot–Vijayaraghavan number or PV number. A PV number is a real algebraic integer, a root of a polynomial equation with integer coefficients, such that all the other roots have absolute value less than one [1]. The set of all PV numbers is infinite and has a remarkable topological structure. The set of all one-dimensional quasi-crystals has a structure at least as rich as the set of all PV numbers and probably much richer. We do not know for sure, but it is likely that a huge universe of one-dimensional quasi-crystals not associated with PV numbers is waiting to be discovered.

Here comes the connection of the onedimensional quasi-crystals with the Riemann hypothesis. If the Riemann hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasi-crystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers. My friend Andrew Odlyzko has published a beautiful computer calculation of the Fourier transform of the zeta-function zeros [2]. The calculation shows precisely the expected structure of the Fourier transform, with a sharp discontinuity at every logarithm of a prime or prime-power number and nowhere else.

My suggestion is the following. Let us pretend that we do not know that the Riemann Hypothesis is true. Let us tackle the problem from the other end. Let us try to obtain a complete enumeration and classification of one-dimensional quasicrystals. That is to say, we enumerate and classify all point distributions that have a discrete point spectrum…We shall then find the well-known quasi-crystals associated with PV numbers, and also a whole universe of other quasicrystals, known and unknown. Among the multitude of other quasi-crystals we search for one corresponding to the Riemann zeta-function and one corresponding to each of the other zeta-functions that resemble the Riemann zeta-function. Suppose that we find one of the quasi-crystals in our enumeration with properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and we can wait for the telephone call announcing the award of the Fields Medal.

These are of course idle dreams. The problem of classifying onedimensional quasi-crystals is horrendously difficult, probably at least as difficult as the problems that Andrew Wiles took seven years to explore. But if we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible. The classification of quasi-crystals is a worthy goal, and might even turn out to be achievable.”

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The 2011 Nobel Prize in chemistry was awarded on Wednesday to an Israeli scientist named Dan Shechtman who discovered a type of crystal so strange and unusual that it upset the prevailing views on the atomic structure of matter, leading to a paradigm shift in chemistry.

But why? What’s so special about quasicrystals?

Schectman’s is an interesting story, involving a fierce battle against established science, ridicule and mockery from colleagues and a boss who found the finding so controversial, he asked him to leave the lab.

Most crystals are composed of a three-dimensional arrangement of atoms that repeat in an orderly pattern. Depending on their chemical composition, they have different symmetries. For example, atoms arranged in repeating cubes have fourfold symmetry. Atoms arranged as equilateral triangles have threefold symmetries. But quasicrystals behave differently than other crystals. They have an orderly pattern that includes pentagons, fivefold shapes, but unlike other crystals, the pattern never repeats itself exactly.

Pat Theil, a senior scientist at the U.S. Energy Department’s Ames Laboratory and a professor of materials science at Iowa State University, uses the analogy of tiling a bathroom floor. Only tiles of certain shapes fit together snugly without creating unsightly holes.

“If you want to cover your bathroom floor, your tiles can be rectangles or triangles or squares or hexagons,” Thiel said. “Any other simple shape won’t work, because it will leave a gap. In a quasicrystal, imagine atoms are at the points of the objects you’re using. What Danny discovered is that pentagonal symmetry works.”

But since pentagons can’t fit together like squares or triangles can, nature places other atomic shapes into the gaps. Glue atoms, she calls them.

Shechtman first saw the startling image while studying a rapidly chilled molten mixture of aluminium and manganese under an electron microscope. It was April 8, 1982. What he saw appeared to counter the laws of nature. He entered it into his notebook, and jotted three question marks next to note.

“Eyn chaya kazo”, Daniel Shechtman said to himself in Hebrew, according to the Royal Swedish Academy of Sciences: “There can be no such creature.”

Thiel, who knows Schectman well, describes him as an “excellent, excellent electron microscopist” who also started a wildly popular entrepreneurship class despite resistance from the university. “If you look at historical materials, people probably had found quasicrystals before,” she said. But they didn’t have the tools or the gumption to say what they were seeing. Danny had both.”

Scientists often study crystals using electron microscopes or X-ray diffraction. By measuring how the X-rays or electrons are diffracted, scientists can determine the patterns in which atoms are arranged inside the crystals.

The finding was more than just conceptual. Quasicrystals have been used in surgical instruments, LED lights and non stick frying pans. They have poor heat conductivity, which makes them good insulators.

“Quasicrystals vary depending on their direction,” said Stephen Nelson, a geology professor who teaches minerology at Tulane University. “One direction might conduct electricity easily – another direction might not conduct electricity at all.”

However, their low tolerance and a restrictive French patent has limited their practical applications, Thiel said.

The real implications of the discovery may still lie in the future, said Bassam Shakhashiri, professor of chemistry at the University of Wisconsin-Madison and president-elect of the American Chemical Society.

“This is how we make progress in science,” he said. “By validating the observations and making sense of them. Subjecting one’s discoveries to scrutiny is very important. It’s a glorious day for scientists, and it demonstrates how important it is to stay in pursuit of knowledge.”

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