Lev Pontryagin

"Pontryagin" redirects here. Not to be confused withPortnyagin.

Lev Pontryagin
Lev Semyonovich Pontryagin (left)
Born	(1908-09-03)3 September 1908 Moscow,Russian Empire
Died	3 May 1988(1988-05-03)(aged 79) Moscow,Soviet Union
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

Lev Semyonovich Pontryagin(Russian:Лев Семёнович Понтрягин,also writtenPontriaginorPontrjagin,first name sometimes anglicized asLeon) (3 September 1908 – 3 May 1988) was aSovietmathematician.Completely blind from the age of 14, he made major discoveries in a number of fields of mathematics, includingalgebraic topology,differential topologyandoptimal control.

Early life and career

He was born inMoscowand lost his eyesight completely due to an unsuccessfuleye surgeryafter aprimus stoveexplosion when he was 14. His mother Tatyana Andreyevna, who did not know mathematical symbols, read mathematical books and papers (notably those ofHeinz Hopf,J. H. C. Whitehead,andHassler 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$\cup$.

In 1925 he enteredMoscow State University,where he was strongly influenced by the lectures ofPavel Alexandrovwho would become his doctoral thesis advisor. After graduating in 1929, he obtained a position atMoscow State University.In 1934 he joined theSteklov Institutein Moscow. In 1970 he became vice president of theInternational Mathematical Union.

Work

Pontryagin worked ondualitytheory forhomologywhile still a student. He went on to lay foundations for the abstract theory of theFourier transform,now calledPontryagin duality.Using these tools, he was able to solve the case ofHilbert's fifth problemfor abelian groups in 1934.

In 1935, he was able to compute the homology groups of the classical compactLie groups,which he would later call his greatest achievement.

WithRené Thom,he is regarded as one of the co-founders ofcobordism theory,and co-discoverers of the central idea of this theory, that framed cobordism andstable homotopyare equivalent.This led to the introduction around 1940 of a theory of certaincharacteristic classes,now calledPontryagin classes,designed to vanish on amanifoldthat is aboundary.

In 1942 he introduced the cohomology operations now calledPontryagin squares.Moreover, inoperator theorythere are specific instances ofKrein spacescalledPontryagin spaces.

Starting in 1952, he worked inoptimal controltheory. Hismaximum principleis fundamental to the modern theory of optimization. He also introduced the idea of abang–bang principle,to describe situations where the applied control at each moment is either the maximum positive 'steer', or the maximum negative 'steer'.

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

Monument to Lev Pontryagin on wall of building onLeninsky AvenueinMoscow,where he lived from 1939 to 1988.

Pontryagin's students includeDmitri Anosov,Vladimir Boltyansky,Revaz Gamkrelidze,Yevgeny Mishchenko,Mikhail Postnikov,Vladimir Rokhlin,andMikhail Zelikin.

Controversy and antisemitism allegations

Pontryagin participated in a few notorious political campaigns in the Soviet Union. In 1930, he and several other young members of theMoscow Mathematical Societypublicly denounced as counter-revolutionary the Society's headDmitri Egorov,who openly supported the Russian Orthodox Church and had recently been arrested. They then proceeded to follow their plan of reorganizing the Society.

Pontryagin was accused of anti-Semitism on several occasions.For example, he attackedNathan Jacobsonfor being a "mediocre scientist" representing the "Zionism movement",while both men were vice-presidents of theInternational Mathematical Union.When a prominent Soviet Jewish mathematician,Grigory Margulis,was selected by theIMUto receive theFields Medalat the upcoming 1978ICM,Pontryagin, who was a member of the executive committee of the IMU at the time, vigorously objected.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.

Pontryagin rejected charges of antisemitism in an article published inSciencein 1979.In his memoirs Pontryagin claims that he struggled withZionism,which he considered a form ofracism.

Publications

• Pontrjagin, L. (1939),Topological Groups,Princeton Mathematical Series, vol. 2, Princeton: Princeton University Press,MR0000265(translated byEmma Lehmer)
• 1952 - Foundations of Combinatorial Topology (translated from 1947 original Russian edition)2015 Dover reprint
• 1962 - Ordinary Differential Equations (translated from Russian by Leonas Kacinskas and Walter B. Counts)
  • Pontryagin, L. S. (15 May 2014).Ordinary Differential Equations: Adiwes International Series in Mathematics.Elsevier.ISBN9781483156491.
• 1962 - withVladimir Boltyansky,Revaz Gamkrelidze,andEvgenii Mishchenko:The Mathematical Theory of Optimal Processes

See also

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

External links

• Lev Pontryaginat theMathematics 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