Wednesday, March 26, 2014

Mathematician rewarded for asking the right questions

I do enjoy reporting for Nature on the Abel Prize in maths, as I’ve done for the past several years. The Norwegians are friendly and helpful – there’s not the absolute secrecy associated with the Nobels, and this is just as well, because more often than not you need a fair bit of advance warning to get your head around what the prize is being given for. This year it was a little less challenging, though, because I already knew a small amount about the Abel laureate Yakov Sinai, whose work is really about physics, even if it demands the most exacting maths. As Sinai put it in the phone conversation through which he was informed of the award “mathematics and physics go together like a horse and carriage” – OK, it’s not exactly a catchy quote, but it is very interesting to see physics formulated with such rigour. I remember hearing years ago how mathematicians generally can’t believe what physicists think is a rigorous argument or proof. But I don’t think they feel that way about Sinai’s work. Anyway, here’s the pre-edit of the Nature story.

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Abel Prize laureate has explored physics “with the soul of a mathematician”

The Norwegian Academy of Science and Letters has awarded the 2014 Abel Prize, often regarded as the “maths Nobel”, to Russian-born mathematical physicist Yakov Sinai of Princeton University. The award cites his “his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics.”

Jordan Ellenberg, a mathematician at the University of Wisconsin who presented the award address today, says that Sinai has worked on questions relating to real physical systems “with the soul of a mathematician”. He has developed tools that show how systems that look superficially different might have deep similarities, much as Isaac Newton showed that the fall of an apple and the movements of the planets are guided by the same principles.

Sinai’s work has been largely in the field now known as complex dynamical systems, which might be regarded as accommodating ideal mechanical laws to the messy complications of the real world. While Newton’s laws of motion provide an approximate description of how objects move under the influence of forces in some simple cases – the motions of the planets, for example – the principles governing real dynamical behaviour are usually more complicated. That’s the case for the weather system and atmospheric flows, population dynamics, physiological processes such as heartbeat, and much else.

Sometimes these movements are subjected to random influences, such as the jiggling of small particles by thermal noise. These are called stochastic dynamical processes. The perfect predictability of Newton’s laws might also be undermined simply by the presence of too many mutually interacting bodies, as in fluid flow. For even just three bodies, Newton’s deterministic laws may lead to chaotic behaviour, meaning that vanishingly small differences in the initial conditions can lead to widely different outcomes over long times. This kind of chaos is now known to be present in the orbits of planets in the solar system.

Sinai has developed mathematical tools for exploring such behaviour. He has identified quantities that remain the same even if the trajectories of objects in these complex dynamical systems become unpredictable. His interest in these issues began while he was at Moscow State University in the late 1950s as a student of Andrey Kolmogorov, one of the greatest mathematical physicists of the twentieth century, who established some of the foundations of probability theory.

Sinai and Kolmogorov showed that even for dynamical systems whose detailed behaviour is unpredictable – whether because of chaos or randomness – there is a quantity that measures just how ‘complex’ the motion is. Inspired by the work of Claude Shannon in the 1940s, who showed that a stream of information can be assigned an entropy, Sinai and Kolmogorov defined a related entropy that measures the predictability of the dynamics: the higher the Kolmogorov-Sinai (K-S) entropy, the lower the predictability.

Ellenberg says that, whereas many physicists might have expected such a measure to distinguish between deterministic systems (where all the interactions are exactly specified) and stochastic ones, the K-S entropy showed that in fact there are qualitatively different types of purely deterministic system: those with zero entropy, which can be predicted exactly, and those with a non-zero entropy which are not wholly predictable, in particular chaotic systems.

Invariant measures like the K-S entropy are related to how thoroughly such a system explores all the different states that it could possibly adopt. A system that ‘visits’ all these states more or less equally on average is said to be ergodic. One of the most important model systems for studying ergodic behaviour is the Sinai billiard, which Sinai introduced in the 1960s. Here a particle bounces around (without losing any energy) within a square perimeter, in the centre of which there is a circular wall. This was the first dynamical system for which it could be proved, by Sinai himself, that all the particle’s trajectories are ergodic – they pass through all of the available space. They are also chaotic, in the sense that the slightest difference in the particle’s initial trajectory leads rather quickly to motions that don’t look at all alike.

In these and other ways, Sinai has laid the groundwork for advances in understanding turbulent fluid flow, the statistical microscopic theory of gases, and chaos in quantum-mechanical systems.

The Abel Prize in mathematics, named after Norwegian mathematician Niels Henrik Abel (1802–29), is modelled on the Nobel prizes and has been awarded every year since 2003. It is worth 6 million Norwegian kroner, or about US$1 million.

“I'm delighted that Sinai, whose scientific and social company I enjoy, has won this prize”, says Michael Berry of Bristol University, who has worked on chaotic quantum billiards and other aspects of complex dynamics.

Ellenberg feels that Sinai’s work has demonstrated how, in maths, “a good definition is as important as a good theorem.” While physicists knew in a loose way what they meant by entropy, he says, Sinai has asked “what are we actually talking about here?” This drive to get the right definition has helped him identify what is truly important and fundamental to the way a system behaves.

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