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Home , Andrey Kolmogorov, Bernt Michael Holmboe, Lennart Carleson, Yakov Sinai

The Laureate 2014

Yakov G. Sinai

Princeton University, USA, and Landau Institute for Theoretical Physics, Russian Academy of Sciences www.abelprize.no Yakov G. Sinai receives the Abel Prize for 2014 “for his fundamental contributions to dynamical systems, , and ” for studying the of a system dynamical systems as well as in the attempts Citation through a -theoretical description to tackle real-life complex chaotic behavior of its trajectories. It has led to very important such as turbulence. advances in the classification of dynamical Sinai’s other pioneering works in systems. mathematical physics include: random Sinai has been at the forefront of walks in a random environment (Sinai’s ergodic theory. He proved the first walks), phase transitions (Pirogov–Sinai theorems for scattering billiards in the style theory), one-dimensional turbulence (the of Boltzmann, work he continued with statistical shock structure of the Bunimovich and Chernov. He constructed Burgers equation, by E–Khanin–Mazel– Markov partitions for systems defined by Sinai), the renormalization group theory iterations of Anosov diffeomorphisms, which (Bleher–Sinai), and the spectrum of discrete led to a series of outstanding works showing Schrödinger operators. the power of to describe Sinai has trained and influenced a various classes of mixing systems. generation of leading specialists in his The Norwegian Academy of Science and Some of these phenomena can be foreseen With Ruelle and Bowen, Sinai discovered research fields. Much of his research has Letters has decided to award the Abel Prize with great accuracy, while others seem to the notion of SRB measures: a rather general become a standard toolbox for mathematical for 2014 to evolve in a chaotic, unpredictable way. Now and distinguished invariant measure for physicists. His works had and continue it has become clear that order and chaos are dissipative systems with chaotic behavior. to have a broad and profound impact on Yakov G. Sinai intimately connected: we may find chaotic This versatile notion has been very useful and physics, as well as on the , USA, and Landau behavior in deterministic systems, and in the qualitative study of some archetypal ever-fruitful interaction of these two fields. Institute for Theoretical Physics, Russian conversely, the statistical analysis of chaotic Academy of Sciences systems may lead to definite predictions. made fundamental “for his fundamental contributions to contributions in this broad domain, dynamical systems, ergodic theory, and discovering surprising connections between mathematical physics” order and chaos and developing the use of and measure theory in the study Ever since the time of Newton, of dynamical systems. His achievements differential equations have been used by include seminal works in ergodic theory, , scientists and engineers to which studies the tendency of a system to explain natural phenomena and to predict explore all of its available states according how they evolve. Many equations incorporate to certain time ; and statistical stochastic terms to model unknown, mechanics, which explores the behavior of seemingly random, factors acting upon that systems composed of a very large number evolution. The range of modern applications of particles, such as molecules in a gas. of deterministic and stochastic evolution Sinai’s first remarkable contribution, equations encompasses such diverse inspired by Kolmogorov, was to develop an issues as planetary motion, ocean currents, invariant of dynamical systems. This invariant physiological cycles, population dynamics, has become known as the Kolmogorov–Sinai Photo: Princeton University, Department of Mathematics and electrical networks, to name just a few. entropy, and it has become a central notion

4 5 The deep contributions made by Sinai London Mathematical Society (1992), the Biography early in his career led to his being invited Hungarian Academy of Sciences (1993), the to lecture at the International Congress of United States National Academy of Sciences Mathematicians in in 1962. Sinai (1999), the Brazilian Academy of Sciences has since been an invited speaker at several (2000), the Academia Europaea (2008), the important international conferences and has Polish Academy of Sciences (2009) and the given many prestigious lectures worldwide. Royal Society of London (2009). He has spoken four times at the International He has received honorary degrees Congress of Mathematicians. He was a from Warsaw University (1993), Budapest Yakov G. Sinai was born on 21 September During 1997-1998 Yakov Sinai was plenary speaker at the 1st Latin American University of Science and Technology (2002), 1935 in , Russia. Both of his parents, Thomas Jones Professor at Princeton Congress in Mathematics in 2000. In 2001 he the Hebrew University in Jerusalem (2005), Gregory Sinai and Nadezda Kagan, were University and in 2005 he was Moore was appointed Chairman of the Fields Medal and Warwick University (2010). microbiologists with research careers. His Distinguished Scholar at the California Institute Committee of International Mathematical Professor Sinai is also respected as a grandfather, the Benjamin of Technology in Pasadena, California. Union, which decided on the awards of the teacher at Princeton. In the words of a former Fedorovich Kagan, was head of the Yakov Sinai is one of the most influential Fields Medals at the Congress in Beijing in student, “It’s quite inspirational to be in his Department of Differential Geometry at mathematicians of the twentieth century. the following year. class … People feel an immediate urge to . Kagan had great He has achieved numerous groundbreaking participate – there is a radiance which comes influence on his grandson. He retired from results in the theory of dynamical systems, Awards and honours from him and inspires us.”2 He is also known his chair at Moscow State University in in mathematical physics and in probability Yakov Sinai has received many distinguished for his persistence in the face of obstacles, be 1952, the year in which his grandson Yakov theory. Many mathematical results are named international awards. In 2013 he was awarded they bureaucratic or theoretical, a trait which Grigorevich entered the Faculty of after him, including Kolmogorov–Sinai the Leroy P. Steele Prize for Lifetime Achieve­ has served him well throughout the years. Mechanics and Mathematics. entropy, Sinai’s billiards, Sinai’s random walk, ment from the American Mathematical — Yakov Sinai received his B.S. (1957), his Sinai-Ruelle-Bowen measures, and Pirogov- Society. Other awards include the Wolf Prize Ph.D. (1960), and his doctorate (1963) from Sinai theory. Sinai is highly respected in in Mathematics (1997), the Nemmers Prize For his 70th birthday in 2005 a special issue Moscow State University. His advisor was both physics and mathematics communities in Mathematics (2002), the Henri Poincaré of the Moscow Mathematical Journal was the prominent . Sinai was as the major architect of the most bridges Prize from the International Association dedicated to Sinai: “Yakov Sinai is one of a Scientific Researcher at the Laboratory connecting the world of deterministic of Mathematical Physics (2009) and the the greatest mathematicians of our time. of Probabilistic and Statistical Methods at (dynamical) systems with the world of Dobrushin International Prize from the Institute The list of international prizes awarded to Moscow State University from 1960 to 1971. probabilistic (stochastic) systems. Perhaps it of Information Transmission of the Russian him as a sign of recognition of his scientific In 1971 he became a Professor at Moscow is only to be expected that he is the author Academy of Sciences (2009). Among his contributions is extremely long, and the list State University and a Senior Researcher at of an article titled “Mathematicians and many recognitions are the Boltzmann Gold of his fundamental results is even longer. 1 the Landau Institute of Theoretical Physics, Physicists = Cats and Dogs?” Medal from the Commission on Statistical His permanent interest in mathematics and Russian Academy of Sciences. Since 1993 During the past half-century Yakov Sinai Physics of the International Union of Pure and his exceptional scientific enthusiasm inspire he has been a Professor of Mathematics has written more than 250 research papers Applied Physics (1986) and the Dirac Medal several generations of scientists all over the at Princeton University, USA, but has and a number of books. Sinai and his wife from the Abdus Salam International Centre for world. His mere presence at a seminar or at concurrently kept his position at the Landau Elena B. Vul, a mathematician and physicist, Theoretical Physics in Trieste (1992). a conference makes scientific life brighter Institute of Theoretical Physics. The Landau have also written a number of joint papers. Many mathematical societies and and more exciting.” Institute was founded in 1964 and is located Yakov Sinai has supervised more than 50 academies have elected Sinai to membership in Chernogolovka, some 40 kilometers Ph.D. students. or honorary membership: the American northeast of Moscow; it is at the heart of 2 Dennis Kosygin, as quoted in The Daily Princetonian, 1 Bulletin (New Series) of the American Mathematical Academy of Arts and Sciences (1983), the a scientific network in the tradition of the Dec. 3, 1996, pp. 1,7, on the occasion of the Society, Vol 43, Number 4, October 2006, pages Russian Academy of Sciences (1991), the awarding of the Wolf Prize. Landau school. 563–565.

6 7 A glimpse of the Laureate’s work In general, we can get a good overview Entropy measures to what extent we are able of what happens in a in to predict the next step in the sequence. the short term. However, when analyzed Another example concerns a container The Entropy of in the long term, dynamical systems are filled with gas. The phase of this difficult to understand and predict. The physical system represents phases of the a Dynamical System problem of weather forecasting illustrates gas, i.e. the position and the momentum of this phenomenon; the weather condition, every single gas molecule, and the laws of Arne B. Sletsjøe described by air pressure, temperature, nature determine the dynamics. Again, the wind, humidity, etc. is a phase of a dynamical degree of complexity and chaotic behaviour system. A weather forecast for the next ten of the gas molecules will be the ingredients in minutes is much more reliable than a weather the concept of entropy. forecast for the next ten days. Summing up, The Kolmogorov–Sinai- Yakov Sinai was the first to come up with entropy measures unpredictability of a In 1948 the American mathematician Claude Towards the end of the 1950s, the a mathematical foundation for quantifying dynamical system. The higher unpredictability, E. Shannon published an article entitled Russian mathematician Andrey Kolmogorov the complexity of a given dynamical system. the higher entropy. This fits nicely with “A Mathematical Theory of Communication”. held a seminar series on dynamical systems Inspired by Shannon’s entropy in information Shannon entropy, where unpredictability His idea was to use the formalism of at Moscow University. A question often raised theory, and in the framework of Kolmogorov’s of the next character is equivalent to new mathematics to describe communication in the seminar concerned the possibility Moscow seminar, Sinai introduced the information. It also fits with the concept of as a phenomenon. The purpose of all of deciding structural similarity between concept of entropy for so-called measure- entropy in thermodynamics, where disorder communication is to convey a message, different dynamical systems. A young seminar preserving dynamical systems, today known increases the entropy, and disorder and but how this is done is the messenger’s participant, Yakov Sinai, presented an as Kolmogorov–Sinai-entropy. This entropy unpredictability are closely related. choice. Some will express themselves using affirmative answer, introducing the concept turned out to be a strong and far-reaching Kolmogorov–Sinai-entropy has numerous words or characters; others prefer of entropy of a dynamical system. invariant of dynamical systems. strongly influenced our understanding to be briefer. The content of the information A dynamical system is a description of a The Kolmogorov–Sinai-entropy provides of the complexity of dynamical systems. is the same, but the information density physical system and its evolution over time. a rich generalization of Shannon entropy. In Even though the formal definition is not may vary. An example is the so-called SMS The system has many phases and all phases a message is an infinite that compli­cated, the concept has shown language. When sending an SMS message are represented in the phase space of the sequence of symbols, corresponding to a its strength through the highly adequate it is common to try to minimize the number system. A path in the phase space describes phase in the framework of dynamical systems. answers to central problems in the of characters. The sentence “I love You” the dynamics of the dynamical system. The shift operator, switching the sequence classification of dynamical systems. consists of 10 characters, while “I <3 U” A dynamical system may be one step, gives the dynamics of the system. consists of only 6, but the content of the two deterministic. In a deterministic system no messages is the same. Shannon introduced is involved in the development the notion of entropy to measure the density of future states of the system. A swinging of information. To what extent does the next pendulum describes a deterministic system. character in the message provide us with Fixing the position and the speed, the laws more information? High Shannon entropy of physics will determine the motion of the means that each new character provides pendulum. When throwing a dice, we have new information; low Shannon entropy the other extreme; a stochastic system. The indicates that the next character just confirms future is completely uncertain, the last toss of something we already know. the dice has no influence on the next.

8 9 About the Abel Prize

The Abel Prize is an international award for Michael Holmboe Memorial Prize for outstanding scientific work in the field of excellence in teaching mathematics in mathematics, including mathematical aspects . In addition, national mathematical of computer science, mathematical physics, contests, and various other projects and probability, numerical analysis, scientific activities are supported in order to stimulate computing, statistics, and also applications of interest in mathematics among children mathematics in the sciences. The Norwegian and youth. Academy of Science and Letters awards the Abel Prize based upon recommendations — from the Abel Committee. The Prize is named after the exceptional Norwegian Call for nominations 2015: mathematician (1802–1829). The Norwegian Academy of Science and According to the statutes of the Abel Prize, Letters hereby calls for nominations for the objective is both to award the annual the Abel Prize 2015, and invite you (or Abel Prize, and to contribute towards raising your society or institution) to nominate the status of mathematics in society and candidate(s). Nominations are confidential stimulating the interest of children and young and a nomination should not be made known people in mathematics. The prize carries a to the nominee. cash award of 6 million NOK (about 750,000 Euro or about 1 million USD) and was first Deadline for nominations for the Abel Prize awarded in 2003. Among initiatives supported 2015 is September 15, 2014. Please consult are the Abel Symposium, the International www.abelprize.no for more information Mathematical Union’s Commission for Developing Countries, the Abel Conference Welcome back in 2015: at the Institute for Mathematics and its Abel Prize Award Ceremony Tuesday May 19, Applications in Minnesota, and The Bernt 2015, University Aula, .

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The Abel Prize Laureates

2014 2013 2012 2011 2010 2009 Yakov G. Sinai Endre Szemerédi John Torrence Tate Mikhail Leonidovich

“for his fundamental contributions to “for seminal contributions to “for his fundamental contributions “for pioneering discoveries in “for his vast and lasting impact Gromov dynamical systems, ergodic theory, algebraic geometry and for their to dis­ crete mathematics and , geometry and algebra.” on the theory of numbers.” “for his revolutionary contributions and mathematical physics.” transformative impact on number theoretical computer science, to geometry.” theory, representation theory, and in recognition of the profound and related fields.” and lasting impact of these contributions on additive number theory and ergodic theory.”

2008 2007 2006 2005 2004 2003 John Griggs Thompson Srinivasa S. R. Varadhan Peter D. Lax Sir Michael Francis Atiyah Jean-Pierre Serre and and Isadore M. Singer “for his fundamental contributions “for his profound and seminal “for his groundbreaking “for playing a key role in shaping “for their profound achievements in to and in contributions to harmonic contributions to the theory and “for their discovery and proof of the the modern form of many parts algebra and in particular for shaping particular for creating a unified analysis and the theory of application of partial differential index theorem, bringing together of mathemat­ics, including modern group theory.” theory of large deviations.” smooth dynamical systems.” equations and to the computation topology, geometry and analysis, topology, algebraic geometry of their solutions.” and their outstand ­ing role in building and number theory.” new bridges between mathematics and theoretical physics.”

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Programme

Abel Week 2014

May 19 Photo front page: Princeton University, Department of Mathematics

Holmboe Prize Award Ceremony Abel Banquet at Akershus Castle in The Minister of Education and Research honour of the Abel Laureate presents the Bernt Michael Holmboe Hosted by the Norwegian Government (by Memorial Prize for teachers of mathematics invitation from the Norwegian Government) at

Wreath-laying ceremony May 21 by the Abel Prize Laureate at the Abel The Abel Lectures Monument in the Palace Park Laureate Lecture, Science Lecture, and other lectures in the field of the Laureate’s work May 20 Georg Sverdrups Hus, Aud. 1, Abel Prize Award Ceremony The Abel Party HRH The Crown Prince presents the Abel at The Norwegian Academy of Science Prize in the University Aula, University of Oslo and Letters Reception and interview with the Abel Laureate May 22 Journalist Tonje Steinsland interviews the Abel Laureate at Det Norske Teatret Laureate Lecture and events for schoolchildren in Stavanger Programme at Vitenfabrikken in Sandnes, and Laureate lecture at the University of Stavanger

The Norwegian Academy Press contact: For other information: of Science and Letters Anne-Marie Astad Trine Gerlyng [email protected] [email protected] +47 22 12 10 92 +47 415 67 406

16 Register online at: www.abelprize.no from mid-April, or contact [email protected]