Lev Semenovich Pontryagin
Encyclopedia
Lev Semenovich Pontryagin (Russian:
Лев Семёнович Понтря́гин) (3 September 1908 – 3 May 1988) was a Soviet
mathematician
. He was born in Moscow
and lost his eyesight due to a primus stove
explosion when he was 14. Despite his blindness he was able to become a mathematician due to the help of his mother Tatyana Andreevna who read mathematical books and papers (notably those of Heinz Hopf
, J. H. C. Whitehead
and Hassler Whitney
) to him. He made major discoveries in a number of fields of mathematics, including algebraic topology
and differential topology
.
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
. In topology he posed the basic problem of cobordism theory. This led to the introduction around 1940 of a theory of certain characteristic class
es, now called Pontryagin class
es, 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 space
s called Pontryagin spaces.
Later in his career he worked in optimal control
theory. His maximum principle is fundamental to the modern theory of optimization. He also introduced there the idea of a bang-bang principle
, to describe situations where either the maximum 'steer' should be applied to a system, or none.
Pontryagin's students include Dmitri Anosov, Vladimir Boltyansky
, Mikhail Postnikov
and Vladimir Rokhlin.
for being a "mediocre scientist" representing "Zionism movement
", while both men were vice-presidents of the International Mathematical Union
. He rejected charges in anti-Semitism in an article published in Science
in 1979, claiming that he struggled with Zionism
which he considered a form of racism
. 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. 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 also participated in a few notorious political campaigns in the Soviet Union, most notably, in the Luzin affair.
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...
Лев Семёнович Понтря́гин) (3 September 1908 – 3 May 1988) was a Soviet
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He was born in Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
and lost his eyesight due to a primus stove
Primus stove
The Primus stove, the first pressurized-burner kerosene stove, was developed in 1892 by Frans Wilhelm Lindqvist, a factory mechanic in Stockholm, Sweden. The stove was based on the design of the hand-held blowtorch; Lindqvist’s patent covered the burner, which was turned upward on the stove...
explosion when he was 14. Despite his blindness he was able to become a mathematician due to the help of his mother Tatyana Andreevna who read mathematical books and papers (notably those of Heinz Hopf
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
, J. H. C. Whitehead
J. H. C. Whitehead
John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....
and Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
) to him. He made major discoveries in a number of fields of mathematics, including algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
and differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
.
Work
He worked on dualityDuality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
theory for homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
while still a student. He went on to lay foundations for the abstract theory of the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, now called Pontryagin duality
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
. In topology he posed the basic problem of cobordism theory. This led to the introduction around 1940 of a theory of certain characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es, now called Pontryagin class
Pontryagin class
In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...
es, designed to vanish on a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
that is a boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
. In 1942 he introduced the cohomology operations now called Pontryagin squares. Moreover, in operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
there are specific instances of Krein space
Krein space
In mathematics, in the field of functional analysis, an indefinite inner product spaceis an infinite-dimensional complex vector space K equipped with both an indefinite inner product\langle \cdot,\,\cdot \rangle \,...
s called Pontryagin spaces.
Later in his career he worked in optimal control
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
theory. His maximum principle is fundamental to the modern theory of optimization. He also introduced there the idea of a bang-bang principle
Bang-bang control
In control theory, a bang–bang controller , also known as a hysteresis controller, is a feedback controller that switches abruptly between two states. These controllers may be realized in terms of any element that provides hysteresis...
, to describe situations where either the maximum 'steer' should be applied to a system, or none.
Vladimir Boltyansky
Vladimir Grigorevich Boltyansky , also transliterated as Boltyanski, Boltyanskii, or Boltjansky, is a Soviet and Russian mathematician, educator and author of popular mathematical books and articles. He is best known for his books on topology, combinatorial geometry and Hilbert's third problem.-...
, Mikhail Postnikov
Mikhail Postnikov
Mikhail Mikhailovich Postnikov was a Soviet mathematician, known for his work in algebraic and differential topology....
and Vladimir Rokhlin.
Controversy and anti-semitism
Pontryagin was a controversial personality. Although he had many Jews among his friends and supported them in his early years, he was accused of anti-Semitism in his mature years. For example he attacked Nathan JacobsonNathan Jacobson
Nathan Jacobson was an American mathematician....
for being a "mediocre scientist" representing "Zionism movement
Zionism
Zionism is a Jewish political movement that, in its broadest sense, has supported the self-determination of the Jewish people in a sovereign Jewish national homeland. Since the establishment of the State of Israel, the Zionist movement continues primarily to advocate on behalf of the Jewish state...
", while both men were vice-presidents of the International Mathematical Union
International Mathematical Union
The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...
. He rejected charges in anti-Semitism in an article published in Science
Science (journal)
Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
in 1979, claiming that he struggled with Zionism
Zionism
Zionism is a Jewish political movement that, in its broadest sense, has supported the self-determination of the Jewish people in a sovereign Jewish national homeland. Since the establishment of the State of Israel, the Zionist movement continues primarily to advocate on behalf of the Jewish state...
which he considered a form of racism
Racism
Racism is the belief that inherent different traits in human racial groups justify discrimination. In the modern English language, the term "racism" is used predominantly as a pejorative epithet. It is applied especially to the practice or advocacy of racial discrimination of a pernicious nature...
. When a prominent Soviet Jewish mathematician, Grigory Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
, was selected by the IMU
International Mathematical Union
The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...
to receive the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
at the upcoming 1978 ICM
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
, Pontryagin, who was a member of the Executive Committee of the IMU
International Mathematical Union
The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...
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 also participated in a few notorious political campaigns in the Soviet Union, most notably, in the Luzin affair.
See also
- Andronov–Pontryagin criterionAndronov–Pontryagin criterionThe Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.-Statement:A dynamical system...
- Pontryagin classPontryagin classIn mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...
- Pontryagin dualityPontryagin dualityIn mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
- Pontryagin's minimum principlePontryagin's minimum principlePontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated by the Russian mathematician Lev Semenovich...
External links
- Autobiography of Pontryagin (in Russian)