Anatoly Borisovich Katok ( August 9, 1944 , Washington, DC) , USA - April 30, 2018 , Danville (Pennsylvania) , USA) - Soviet and American mathematician, specialist in the theory of dynamical systems , director of the Center for Dynamical Systems and Geometry at the University of Pennsylvania .
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| Scientific field | mathematician |
| Place of work | Moscow State University , University of Maryland College Park , California Institute of Technology , Pennsylvania State University |
| Alma mater | MSU (mehmat) |
| supervisor | J. G. Sinai |
| Famous students | Michael Brin |
| Known as | specialist in the theory of dynamical systems |
| Awards and prizes | [d] |
Early life and education
Parents - Berl (Boris) Lazarevich Katok (1901-1963), metallurgical engineer, and Dweira (Dora) Sorkin (1905-1998) [2] . He studied at three Moscow schools: 69 (first and eighth grades), 637 (second to seventh) and 59 (ninth grade). For the tenth grade, he passed exams externally and entered the Moscow State University. He graduated from the Faculty of Mechanics and Mathematics of Moscow State University in 1965 and graduate school in 1968, received a Ph.D. in physics and mathematics, defending a dissertation entitled “Application of the method of approximation of dynamical systems by periodic transformations to ergodic theory” under the guidance of Yakov Sinai .
In 1978, with his wife, mathematician Svetlana Katok , he emigrated to the United States.
Work and research
Back in graduate school, together with Anatoly Styopin, he developed the theory of periodic approximations preserving the measure of transformations, known as the Katka - Stepin approximations. This theory helped solve classical problems that go back to von Neumann and Kolmogorov , in 1967 he received the prize of the Moscow Mathematical Society for this work. The next major result is the theory of monotone equivalence (known as the Kakutani equivalence theory), which is based on a far-reaching generalization of the concept of time replacement for flows.
In the theory of dynamical systems, Katk owns numerous constructions that reveal interesting and often unexpected phenomena. Among them are the Anosov-Katka construction of smooth ergodic measure-preserving diffeomorphisms on compact manifolds, the Bernoulli diffeomorphism construction with nonzero Lyapunov exponents on any surface, and the first known construction of the invariant foliation for which Fubini's theorem does not hold ( English Fubini foiled ).
In a joint work with Lindenstrauss and Einsiedler ( German: Manfred Einsiedler ), progress was made in the search for evidence of the Littlewood hypothesis in the theory of Diophantine approximations .
He formulated a number of hypotheses and problems that influenced the development of the theory of dynamical systems. The most famous of them is the Rink hypothesis of entropy, which connects the important geometric and dynamic properties of systems in one formula, which is one of the first statements in the theory of rigidity of dynamical systems, which was connected with the main works of the scientist in the 2000s - 2010s. Among his works are studies of the rigidity of the smooth actions of Abelian groups and lattices in senior Lie groups .
One of Katka’s most famous and widely cited works is on the topological properties of non-uniformly hyperbolic dynamical systems, which describes the density of periodic points and gives lower bounds on their number, and also gives an approximation of the topological entropy of the system by its values on hyperbolic sets (horseshoes). These topics were addressed in his invited paper at the International Congress of Mathematicians in 1983 (“Unequal Hyperbolicity and the Structure of Smooth Dynamical Systems”), as well as the Rufus Bowen Memorial Lecture at the University of California at Berkeley in 1982.
Together with his former student Boris Hasselblatt ( German: Boris Hasselblatt ), he wrote the book Introduction to the Modern Theory of Dynamical Systems ( Cambridge University Press , 1995).
He was the chief editor of the Journal of Modern Dynamics , one of the editors of Ergodic Theory and Dynamical Systems , and a member of the editorial boards of many publications, including Cambridge Tracts in Mathematics and Cambridge Studies in Advanced Mathematics .
Teaching
He was a professor of mathematics at the University of Maryland at College Park (1978-1984) and at the California Institute of Technology (1984-1990). Since 1990, he has been a professor at the University of Pennsylvania. Graduated 37 doctoral students.
In 1996, together with his wife Svetlana Katok and Anatoly Kushnirenko, he organized a special MASS program, which selected undergraduate students from various universities study for one semester at the University of Pennsylvania.
Awards and recognition
In 1967, together with Styopin and Oseledets received the prize of the Moscow Mathematical Society .
He has been a member of the American Academy of Arts and Sciences since 2004. Since 2012 he was a full member of the American Mathematical Society [3] .
Selected Monographs
- A. B. Katok, B. Hasselblatt. Introduction to the modern theory of dynamical systems / Per. from English A. Kononenko with the participation of S. Ferleger. - M .: Factorial, 1999 .-- 768 p. - ISBN 5-88688-042-9 .
- Katok A. B., Hasselblatt B. Introduction to the modern theory of dynamical systems with a review of recent achievements / trans. from English under the editorship of A. S. Gorodetsky. - M .: ICMMO, 2005 .-- 464 p. - ISBN 5-94057-063-1 .
- Anatole Katok. Combinatorial Constructions in Ergodic Theory and Dynamics . - Pennsylvania State University, University Park, PA, 2003. - (University Lecture Series). - ISBN 978-0-8218-3496-1 .
- Handbook of Dynamical Systems / Anatole Katok, Boris Hasselblatt (eds.). - Elseveir, 2002 .-- Vol. 1A. - ISBN 0-444-82669-6 .
- Handbook of Dynamical Systems / Anatole Katok, Boris Hasselblatt (eds.). - Elseveir, 2005 .-- Vol. 1B. - ISBN 0-444-52055-4 .
- Anatole Katok, Vaughn Climenhaga. Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them . - Pennsylvania State University - AMS, 2008. - ISBN 978-0-8218-4679-7 .
- Anatole Katok, Viorel Nitica. Rigidity in Higher Rank Abelian Group Actions . - Cambridge University Press, June 2011. - Vol. 1.- ISBN 9780521879095 . Archived on February 28, 2014. Archived February 28, 2014 on Wayback Machine
Notes
- ↑ 1 2 German National Library , Berlin State Library , Bavarian State Library , etc. Record # 111187125 // General regulatory control (GND) - 2012—2016.
- ↑ Anatole Katok . Geni.com Date of treatment July 16, 2015.
- ↑ List of Fellows of the American Mathematical Society . Date of treatment October 27, 2013.
Literature
- MI Brin, B. Hasselblat, Yu. S. Ilyashenko, AG Kushnirenko, Ya. B. Pesin, AB Sossinski, MA Tsfasman. Anatole Katok // Mosc. Math. J. - 2004. - Vol. 4: 4. - P. 977–979.
- A. Venkatesh . The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture // Bull. AMS. - 2008. - Vol. 45. - P. 117–134.
- Modern Dynamical Systems and Applications / Michael Brin, Boris Hasselblatt, Yakov Pesin (Eds.). - Cambridge University Press, 2005. - ISBN 9780521840736 .
- Eugene B. Dynkin . Collection of Mathematical Interviews . Date of treatment July 16, 2015.