# Yakov Grigorevich Sinai

### Born: 21 September 1935 in Moscow, Russia

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**Yakov Grigorevich Sinai**'s parents, Gregory Sinai and Nadezda Kagan, were microbiologists with research careers. However the family had strong mathematical connections since Yakov Grigorevich's grandfather (Nadezda Kagan's father) was Benjamin Fedorovich Kagan, the Head of the Department of Differential Geometry at Moscow State University where he founded an important School of Differential Geometry. It is also worth recording that the family was Jewish, and Kagan had a long struggle against anti-Semitism. It was, however, a family which had over several generations taken a leading role in Russian scientific and cultural life. Kagan had a large influence on his grandson. He retired from his chair at Moscow State University in 1952, the year in which his grandson Yakov Grigorevich entered the Faculty of Mechanics and Mathematics.

Sinai's first advisor at Moscow State University was Nikolai Guryevich Chetaev who was an expert on analytical mechanics, particularly on stability of motion. Sinai quickly became interested in the dynamical systems on which Chetaev worked. However, he changed advisors and began to work with Evgenii Borisovich Dynkin. The problem which Dynkin suggested that Sinai work on, led to his first paper

*On the distribution of the first positive sum for a sequence of independent random variables*(Russian) (1957). In 1957 Sinai was awarded his first degree from Moscow State University and began to undertake research for his Master's Degree (equivalent to a Ph.D.) working with Andrei Nikolaevich Kolmogorov. He was awarded this degree in 1960 and, in the same year, appointed as a Scientific Researcher at the Laboratory of Probabilistic and Statistical Methods at Moscow State University. He continued to work towards his doctorate (equivalent to the German habilitation) under Kolmogorov. Other members of staff had a major influence on him at this time, particularly Israil Moiseevic Gelfand and Vladimir Abramovich Rokhlin who led the seminar on the metric theory of dynamical systems. Sinai's papers published around the time he was working for his Master's Degree include:

*On the concept of entropy for a dynamic system*(Russian) (1959);

*Flows with finite entropy*(1959) (Russian);

*The central limit theorem for geodesic flows on manifolds of constant negative curvature*(1960) (Russian); and

*Dynamical systems and stationary Markov processes*(1960) (Russian).

Already, in the first of these 1959 papers, Sinai gives theorems which make it possible to calculate the entropy for a large variety of dynamical systems. The term 'Kolmogorov-Sinai entropy' was quickly established [10]:-

The high quality and importance of Sinai's papers led to him being invited to lecture at the International Congress of Mathematicians in Stockholm in 1962. Dynkin and Gelfand were both invited plenary speakers but did not attend. Kolmogorov did attend the Congress and read Dynkin's lecture. In 1971, following Sergei Petrovich Novikov's advice, Sinai accepted a position as Senior Researcher at the L D Landau Institute of Theoretical Physics of the USSR Academy of Sciences. Novikov had just been appointed as head of the Mathematics Division at the Institute. Sinai continued to teach at Moscow State University but he did not become a professor there until 1981. The authors of [9] explain the reasons:-Sinai's work deals with measuring dynamical systems, or systems that change over time, such as weather, the motion of planets and economic systems. These systems can be accurately measured in the short term(short term being relative to the issue at hand); but when analyzed in the long term, the systems are difficult to understand and predict. Sinai was the first to come up with a mathematical foundation for determining the number that defines the complexity of a given dynamical system. His mathematical system is called Kolmogorov-Sinai entropy.

Alexander Sergeyevich Esenin-Volpin was both a poet and a mathematician who led a human rights movement in the Soviet Union. Beginning in 1949, Volpin spent many years in prison for anti-Soviet poetry or in exile as a socially dangerous person. Sinai suffered much for his support of Volpin. For example in 1970 he was invited to lecture at the International Congress of Mathematicians in Nice. However, he was not allowed to go to Nice by the Soviet authorities. Many others were also prevented from attending the Nice Congress including Dynkin, Gelfand, Linnik, Manin, Shafarevich and Sergei Novikov who should have received a Fields Medal at the Congress. Sinai was able to accept the invitation, however, to deliver one of the plenary lectures at the International Congress of Mathematicians in Kyoto in 1990; he spoke onHis signing(together with many other mathematicians)in1968of the well-known letter in defence of A S Esenin-Vol'pin was for a long time a barrier preventing his becoming a Professor(he became a Professor only in1981,17years after submitting his Ph.D. thesis).

*Hyperbolic Billiards*. Continuing with his contributions to the International Congress of Mathematicians, in 2001 he was appointed Chairman of the Fields Medal Committee of International Mathematical Union which decided on the awards of the Fields Medals at the Congress in Beijing in the following year.

In 1993 he was appointed Professor in the Department of Mathematics at Princeton University. He continued with his appointment the L D Landau Institute of Theoretical Physics but gave up his position at Moscow State University. During 1997-1998 he was Thomas Jones Professor of Princeton University and in 2005 he was Moore Distinguished Scholar at the California Institute of Technology at Pasadena, California. He continues to hold his professorship at Princeton University and at the L D Landau Institute of Theoretical Physics in Moscow.

We have already looked the deep contribution made by which Sinai early in his career. Perhaps the best summary of his achievements is given in [3]:-

Sinai has received many major awards, prizes and honours for his remarkable contributions. For example he has received the following medals and prizes:Sinai has done foundational, deep and highly influential work in the fields of ergodic theory, dynamical systems and statistical mechanics. Already in the sixties he had a deep understanding of the principles of what is now called chaos, and was among the first to recognise the significance of this phenomenon for dynamics. He has also done fundamental work in statistical mechanics. Besides his many major contributions to these subjects, he has had a very wide influence through a number of well-known expository texts and through his many research students.

Sinai's work centres round the grand aim of deriving the basic physical laws which describe the behaviour of many particle systems as a direct consequence of simple rules governing the interaction of individual particles. In this he has had some remarkable successes. In ergodic theory his work on hyperbolic systems, on billiards and the hard sphere gas has laid the foundation of many of the techniques presently used for proving that such systems are ergodic and for studying the finer statistical properties of their behaviour. He has influenced the general trend of ergodic theory away from the study of rather artificially constructed examples back to the problem which originally motivated the subject, namely the substantiation of Boltzmann's ergodic hypothesis.

The idea of applying the Kolmogorov theory of entropy to smooth dynamical systems was Sinai's. Previous work of the Russian school had studied entropy, as introduced by Kolmogorov, entirely in the context of probabilistic systems. His results in this direction were new and unexpected. He investigated the class of dynamical systems with transversal foliations, now known as stable and unstable manifolds, and proved that all systems in this class were ergodic, mixing and K. Subsequently he introduced the idea of Markov partitions and constructed such partitions for hyperbolic systems.

Sinai laid the foundations of the theory of billiards, for which he has more recently also constructed Markov partitions, and of the motion of a hard sphere gas. He has made many contributions to statistical mechanics, in particular to the theory of phase transition. His book on this topic is well known. In recent years, Sinai has made important contributions to KAM theory using renormalisation methods. He is currently developing some entirely new and very interesting ideas in quantum chaos.

the Boltzmann Gold Medal from the Commission on Statistical Physics of the International Union of Pure and Applied Physics (1986); the Heineman Prize from the American Physical Society (1989); the Markov Prize from the USSR Academy of Sciences (1990); the Dirac Medal from the Abdus Salam International Centre for Theoretical Physics in Trieste (1992); the Wolf Prize in Mathematics (1997); the Brazilian Award of Merits in Sciences (2000); the Moser Prize from the Society for Industrial and Applied Mathematics (2001); the Frederic Esser Nemmers Prize in Mathematics (2002); the Kolmogorov Lecture and Medal, University of London (2007); the Lagrange Prize from the Institute for Scientific Interchange, Torino, Italy (2008); the Henri Poincaré Prize from the International Association of Mathematical Physics (2009); the Dobrushin International Prize from the Institute of Information Transmission of the USSR Academy of Sciences (2009) and the Abel Prize (2014).

Here are some extracts from the citations for these awards. The Wolf Prize (1997) [5]:-

The 2002 Frederic Esser Nemmers Prize in Mathematics [10]:-Sinai received the prize for "his fundamental contributions to mathematically rigorous methods in statistical mechanics and the ergodic theory of dynamical systems and their applications in physics." Sinai brings to bear on the problems of mathematical physics the powerful tools of dynamical systems and probability theory, often developing new tools for this purpose. He is generally recognized as the world leader in the mathematics of statistical physics. Working in the tradition of the Kolmogorov school, he first formulated the rigorous definition of the invariant entropy for an arbitrary measure-preserving map. His subsequent work covers areas from the ergodicity of the motion of billiards to spectral properties of quasi-periodic Schrödinger operators. Statistical mechanics is one of the most active and rewarding areas of modern mathematics, and Yakov Sinai is its recognized leader today.

The Henri Poincaré Prize (2009) was awarded to Sinai:-His work has revolutionized the study of dynamical systems and influenced statistical mechanics, probability theory and statistical physics.

Yakov Sinai won the 2014 Abel Prize:-... for his ground-breaking works concerning dynamical entropy, ergodic theory, chaotic dynamical systems, microscopic theory of phase transitions, and time evolution in statistical mechanics.

Many mathematical societies and academies have elected Sinai to membership or honorary membership: the American Academy of Arts and Sciences (1983); the USSR Academy of Sciences (1991); the London Mathematical Society (1992); the Hungarian Academy of Sciences (1993); the United States National Academy of Sciences (1999); the Brazilian Academy of Sciences (2000); the Academia Europaea (2008); and the Royal Society of London (2009). He has received honorary degrees from: Warsaw University (1993); Budapest University of Science and Technology (2002); and the Hebrew University in Jerusalem (2005).... for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics.

Sinai has also been invited to give many prestigious lectures or lecture courses including: Loeb Lecturer, Harvard University (1978); Plenary Speaker at the International Congress on Mathematical Physics in Berlin (1981); Plenary Speaker at the International Congress on Mathematical Physics in Marseilles (1986); Distinguished Lecturer, Israel (1989); Solomon Lefschetz Lectures, Mexico (1990); Plenary Speaker at the International Congress of Mathematicians, Kyoto (1990); Landau Lectures, Hebrew University of Jerusalem (1993); Plenary Speaker at the First Latin American Congress in Mathematics (2000); Plenary Speaker at the American Mathematical Society Meeting "Challenges in Mathematics" (2000); Andreevski Lectures, Berlin, Germany (2001); Bowen Lectures, University of California at Berkeley (2001); Leonidas Alaoglu Memorial Lecture, California Institute of Technology (2002); Joseph Fels Ritt Lectures, Columbia University (2004); Leonardo da Vinci Lecture, Milan, Italy (2006); Galileo Chair, Pisa, Italy (2006); John T Lewis Lecture Series, Dublin Institute for Advanced Studies and the Hamilton Mathematics Institute, Trinity College, Dublin, Ireland (2007); and Milton Brockett Porter Lecture Series, Rice University, Houston, Texas (2007).

For his seventieth birthday in 2005 a special issue of the Moscow Mathematical Journal was dedicated to Sinai:-

Yakov Grigorievich Sinai is married to Elen B Vul who is a mathematician and physicist. They have written a number of joint papers. They have one son.Yakov Grigorievich Sinai is one of the greatest mathematician of our days. The list of international prizes awarded to him as a sign of recognition of his scientific contributions is extremely long, the list of his fundamental results being even longer. His permanent interest in mathematics and his exceptional scientific enthusiasm inspires several generations of scientists all over the world. His mere presence at a seminar or at a conference makes scientific life brighter and more exciting.

**Article by:** *J J O'Connor* and *E F Robertson*

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**List of References** (10 books/articles)

**A Poster of Yakov Grigorevich Sinai**

**Mathematicians born in the same country**

**Honours awarded to Yakov Grigorevich Sinai**

(Click the link below for those honoured in this way)

1. | Wolf Prize | 1996/7 | |

2. | Bowen Lecturer | 2001-02 | |

3. | International Congress Speaker | 1990 | |

4. | LMS Honorary Member | 1992 | |

5. | MAA Hedrick Lecturer | 2000 | |

6. | Nemmers Prize | 2002 | |

7. | AMS Steele Prize | 2013 | |

8. | Abel Prize | 2014 |

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