Yakov Sinai. Photo © Mathematisches Forschungsinstitut Oberwolfach, CC-BY-SA 2.0
The Abel Prize for 2014 has gone to Yakov Sinai of Princeton University, “for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics”.
The announcement, made in Norway this morning, was accompanied by an explanation of Sinai’s work, presented by Jordan Ellenberg. You can watch that at the Abel Prize live stream page.
There are also some oldy-worldy fuddy-duddy text explanations written by Arne B. Sletsjøe:
It would be ridiculous to expect us to do a better job of explaining who Sinai is than the committee who have spent so long picking him, wouldn’t it? So, here’s the text of their citation justifying their choice:
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.
Sinai’s first remarkable contribution, inspired by Kolmogorov, was to develop an invariant of dynamical systems. This invariant has become known as the Kolmogorov–Sinai entropy, and it has become a central notion for studying the complexity of a system through a measure-theoretical description of its trajectories. It has led to very important advances in the classification of dynamical systems.
Sinai has been at the forefront of ergodic theory. He proved the first ergodicity theorems for scattering billiards in the style of Boltzmann, work he continued with Bunimovich and Chernov. He constructed Markov partitions for systems defined by iterations of Anosov diffeomorphisms, which led to a series of outstanding works showing the power of symbolic dynamics to describe various classes
of mixing systems.With Ruelle and Bowen, Sinai discovered the notion of SRB measures: a rather general and distinguished invariant measure for dissipative systems with chaotic behavior. This versatile notion has been very useful in the qualitative study of some archetypal dynamical systems as well as in the attempts to tackle real-life complex chaotic behavior such as turbulence.
Sinai’s other pioneering works in mathematical physics include: random walks in a random environment (Sinai’s walks), phase transitions (Pirogov–Sinai theory), one-dimensional turbulence (the statistical shock structure of the stochastic Burgers equation, by E–Khanin–Mazel–Sinai), the renormalization group theory (Bleher–Sinai), and the spectrum of discrete Schrödinger operators.
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.
His works had and continue to have a broad and profound impact on mathematics and physics, as well as on the ever-fruitful interaction of these two fields.
More information
Russian mathematician receives the 2014 Abel Prize news item on the Abel Prize site.
Live stream (now a recorded stream) of the award announcement and explanation by Jordan Ellenberg
Previous Abel laureates, covered here: