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**Lev Semyonovich Pontryagin** (Russian: Лев Семёнович Понтрягин, also written **Pontriagin** or **Pontrjagin**, first name sometimes anglicized as **Leon**) (3 September 1908 – 3 May 1988) was a Soviet mathematician. Completely blind from the age of 14, he made major discoveries in a number of fields of mathematics, including algebraic topology, differential topology and optimal control.

Lev Pontryagin | |
---|---|

Born | |

Died | 3 May 1988 Moscow, Soviet Union | (aged 79)

Nationality | Soviet Union |

Known for | Pontryagin duality Pontryagin class Pontryagin cohomology operation Pontryagin's maximum principle Andronov–Pontryagin criterion |

Scientific career | |

Fields | Mathematics |

Doctoral advisor | Pavel Alexandrov |

Doctoral students | Dmitri Anosov Vladimir Boltyansky Revaz Gamkrelidze Mikhail Postnikov Mikhail Zelikin |

He was born in Moscow and lost his eyesight completely due to an unsuccessful eye surgery after a primus stove explosion when he was 14. His mother Tatyana Andreyevna, who did not know mathematical symbols, read mathematical books and papers (notably those of Heinz Hopf, J. H. C. Whitehead, and Hassler Whitney) to him, and later worked as his secretary. His mother used alternative names for math symbols, such as "tails up" for the set-union symbol .^{[1]}

In 1925 he entered Moscow State University, where he was strongly influenced by the lectures of Pavel Alexandrov who would become his doctoral thesis advisor. After graduating in 1929, he obtained a position at Moscow State University. In 1934 he joined the Steklov Institute in Moscow. In 1970 he became vice president of the International Mathematical Union.

Pontryagin worked on duality theory for homology while still a student. He went on to lay foundations for the abstract theory of the Fourier transform, now called Pontryagin duality. Using these tools, he was able to solve the case of Hilbert's fifth problem for abelian groups in 1934.

In 1935, he was able to compute the homology groups of the classical compact Lie groups, which he would later call his greatest achievement.^{[2]}

With René Thom, he is regarded as one of the co-founders of cobordism theory, and co-discoverers of the central idea of this theory, that framed cobordism and stable homotopy are equivalent.^{[3]} This led to the introduction around 1940 of a theory of certain characteristic classes, now called Pontryagin classes, designed to vanish on a manifold that is a boundary.

In 1942 he introduced the cohomology operations now called Pontryagin squares. Moreover, in operator theory there are specific instances of Krein spaces called Pontryagin spaces.

Starting in 1952, he worked in optimal control theory. His maximum principle is fundamental to the modern theory of optimization. He also introduced the idea of a bang–bang principle, to describe situations where the applied control at each moment is either the maximum positive 'steer', or the maximum negative 'steer'.^{[citation needed]}

Pontryagin authored several influential monographs as well as popular textbooks in mathematics.

Pontryagin's students include Dmitri Anosov, Vladimir Boltyansky, Revaz Gamkrelidze, Yevgeny Mishchenko, Mikhail Postnikov, Vladimir Rokhlin, and Mikhail Zelikin.

Pontryagin participated in a few notorious political campaigns in the Soviet Union. In 1930, he and several other young members of the Moscow Mathematical Society publicly denounced as counter-revolutionary the Society's head Dmitri Egorov, who openly supported the Russian Orthodox Church and had recently been arrested. They then proceeded to follow their plan of reorganizing the Society.^{[2]}

Pontryagin was accused of anti-Semitism on several occasions.^{[2]} For example, he attacked Nathan Jacobson for being a "mediocre scientist" representing the "Zionism movement", while both men were vice-presidents of the International Mathematical Union.^{[4]}^{[5]} When a prominent Soviet Jewish mathematician, Grigory Margulis, was selected by the IMU to receive the Fields Medal at the upcoming 1978 ICM, Pontryagin, who was a member of the executive committee of the IMU at the time, vigorously objected.^{[6]} Although the IMU stood by its decision to award Margulis the Fields Medal, Margulis was denied a Soviet exit visa by the Soviet authorities and was unable to attend the 1978 ICM in person.^{[6]}

Pontryagin rejected charges of antisemitism in an article published in *Science* in 1979,^{[7]} claiming that he struggled with Zionism, which he considered a form of racism.^{[5]}

- Pontrjagin, L. (1939),
*Topological Groups*, Princeton Mathematical Series, vol. 2, Princeton: Princeton University Press, MR 0000265 (translated by Emma Lehmer)^{[8]} - 1952 - Foundations of Combinatorial Topology (translated from 1947 original Russian edition)
^{[9]}2015 Dover reprint^{[10]} - 1962 - Ordinary Differential Equations (translated from Russian by Leonas Kacinskas and Walter B. Counts)
^{[11]}- Pontryagin, L. S. (15 May 2014).
*Ordinary Differential Equations: Adiwes International Series in Mathematics*. Elsevier. ISBN 9781483156491.

- Pontryagin, L. S. (15 May 2014).
- 1962 - with Vladimir Boltyansky, Revaz Gamkrelidze, and Evgenii Mishchenko : The Mathematical Theory of Optimal Processes
^{[12]}

- Andronov–Pontryagin criterion for planar dynamical systems
- Kuratowski's theorem, also called the Pontryagin–Kuratowski theorem, on planar graphs
- Pontryagin class
- Pontryagin duality
- Pontryagin's maximum principle

**^**Robertson, Edmund; O'Connor, John (1999). "Lev Pontryagin - Biography".*MacTutor*. Retrieved 2023-06-19.- ^
^{a}^{b}^{c}Maritz, Pieter (2003-06-01). "Around the graves of Petrovskii and Pontryagin".*The Mathematical Intelligencer*.**25**(2): 55–73. doi:10.1007/BF02984835. ISSN 0343-6993. S2CID 122503334. **^**Mackenzie, Dana (2010),*What's Happening in the Mathematical Sciences, Volume 8*, American Mathematical Society, p. 126, ISBN 9780821849996.**^**O'Connor, John J; Edmund F. Robertson "Nathan Jacobson". MacTutor History of Mathematics archive.- ^
^{a}^{b}Memoirs, by Lev Pontryagin, Narod Publications, Moscow, 1998 (in Russian). - ^
^{a}^{b}Olli Lehto. Mathematics without borders: a history of the International Mathematical Union. Springer-Verlag, 1998. ISBN 0-387-98358-9; pp. 205-206 **^**Pontryagin, LS (September 14, 1979). "Soviet Anti-Semitism: Reply by Pontryagin".*Science*.**205**(4411): 1083–1084. Bibcode:1979Sci...205.1083P. doi:10.1126/science.205.4411.1083. PMID 17735029.**^**Puckett Jr, W. T. (1940). "Book Review:*Topological Groups*".*Bulletin of the American Mathematical Society*.**46**(5): 382–385. doi:10.1090/S0002-9904-1940-07199-X.**^**Massey, W. S. (1953). "Book Review:*Foundations of combinatorial topology*".*Bulletin of the American Mathematical Society*.**59**(2): 188–190. doi:10.1090/S0002-9904-1953-09702-6.**^**Satzer, William J. (October 21, 2015). "Review of*Foundations of combinatorial topology*".*MAA Reviews, Mathematical Association of America*.**^**Brauer, Fred (June 1964). "Review of*Ordinary Differential Equations*by L. S. Pontryagin".*Canadian Mathematical Bulletin*.**7**(2): 315–316. doi:10.1017/S0008439500027119. page 316**^**Blum, Edward Kenneth (December 1963). "Reviewed Work:*The Mathematical Theory of Optimal Processes*by Pontryagin, Boltyanskii, Gamkrelidze, Mishchenko".*The American Mathematical Monthly*.**70**(10): 1114–1116. doi:10.2307/2312867. JSTOR 2312867.

- Lev Pontryagin at the Mathematics Genealogy Project
- Autobiography of Pontryagin (in Russian)
- Kutateladze S. S., Sic Transit... or Heroes, Villains, and Rights of Memory.
- Kutateladze S. S., The Tragedy of Mathematics in Russia