## 2014 Abel Prize

The Norwegian Academy of Science and Letters this morning announced the recipient of its 2014 Abel Prize, the twelfth one awarded. With mathematicians so rarely in the news, I have made it a point here at Ron’s View each year to write a post about the award. (Click on the following links for 2009, 2010, 2011, 2012, and 2013.) This year’s recipient is Yakov G. Sinai, a professor of mathematics at Princeton University and researcher at the Landau Institute of Theoretical Physics outside Moscow.

As I explain each year, the Abel Prize was established in 2001 by the Norwegian government to be the counterpart in mathematics to the Nobel Prizes in other disciplines. It has been awarded by the Norwegian Academy of Science and Letters each year since 2003 to one or two outstanding mathematicians and honors the great, early-nineteenth-century Norwegian mathematician Niels Abel.

Regarding Sinai, here is a passage from the press release:

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2014 to Yakov G. Sinai (78) of Princeton University, USA, and the Landau Institute for Theoretical Physics, Russian Academy of Sciences, “

for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics”.[snip]

Yakov Sinai is one of the most influential mathematicians of the twentieth century. He has achieved numerous groundbreaking results in the theory of dynamical systems, in mathematical physics and in probability theory. Many mathematical results are

named after him … .Sinai is highly respected in both physics and mathematics communities as the major architect of the most bridges connecting the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems. During the past half-century Yakov Sinai has written more than 250 research papers and a number of books. He has supervised more than 50 Ph.D.-students.

Yakov Sinai has trained and influenced a generation of leading specialists in his research fields. Much of his research has become a standard toolbox for mathematical physicists. The Abel Committee says, “His works had and continue to have a broad and profound impact on mathematics and physics, as well as on the ever-fruitful interaction between these two fields.”

The Abel Prize website has a brochure on this year’s prize that is a bit more informative about Sinai’s work and career. Here are the first two paragraphs of an expanded description of his work.

Ever since the time of Newton, differential equations have been used by mathematicians, scientists and engineers to explain natural phenomena and to predict how they evolve. Many equations incorporate stochastic terms to model unknown, seemingly random, factors acting upon that evolution. The range of modern applications of deterministic and stochastic evolution equations encompasses such diverse issues as planetary motion, ocean currents, physiological cycles, population dynamics, and electrical networks, to name just a few. Some of these phenomena can be foreseen with great accuracy, while others seem to evolve in a chaotic, unpredictable way. Now it has become clear that order and chaos are intimately connected: we may find chaotic behavior in deterministic systems, and conversely, the statistical analysis of chaotic systems may lead to definite predictions.

Yakov Sinai made fundamental contributions in this broad domain, discovering surprising connections between order and chaos and developing the use of probability and measure theory in the study of dynamical systems. His achievements include seminal works in ergodic theory, which studies the tendency of a system to explore all of its available states according to certain time statistics; and statistical mechanics, which explores the behavior of systems composed of a very large number of particles, such as molecules in a gas.

See also the short article by Arne Sletsjøe (Norwegian mathematician and former Olympic sprint canoer) at the end of the brochure in which he explains what dynamical systems and entropy are, leading to a glimpse into Sinai’s contributions. I’ll close with an excerpt.

A dynamical system is a description of a physical system and its evolution over time. The system has many phases and all phases are represented in the phase space of the system. A path in the phase space describes the dynamics of the dynamical system.

A dynamical system may be deterministic. In a deterministic system no randomness is involved in the development of future states of the system. A swinging pendulum describes a deterministic system. Fixing the position and the speed, the laws of physics will determine the motion of the pendulum. When throwing a dice, we have the other extreme; a stochastic system. The future is completely uncertain, the last toss of the dice has no influence on the next.

In general, we can get a good overview of what happens in a dynamical system in the short term. However, when analyzed in the long term, dynamical systems are difficult to understand and predict. The problem of weather forecasting illustrates this phenomenon; the weather condition, described by air pressure, temperature, wind, humidity, etc. is a phase of a dynamical system. A weather forecast for the next ten minutes is much more reliable than a weather forecast for the next ten days.

Yakov Sinai was the first to come up with a mathematical foundation for quantifying

the complexity of a given dynamical system. Inspired by Shannon’s entropy in information theory, and in the framework of Kolmogorov’s Moscow seminar, Sinai introduced the concept of entropy for so-called measure-preserving dynamical systems, today known as Kolmogorov–Sinai-entropy. This entropy turned out to be a strong and far-reaching invariant of dynamical systems.

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