Lev Semyonovich Pontryagin (, also written Pontriagin or Pontrjagin, first name sometimes anglicized as Leon) (3 September 1908 – 3 May 1988) was a

Soviet The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. Completely blind from the age of 14, he made major discoveries in a number of fields of mathematics, including

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

and

optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...

Early life and career

He was born in

Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...

and lost his eyesight completely due to an unsuccessful

eye surgery Eye surgery, also known as ophthalmic surgery or ocular surgery, is surgery performed on the eye or its adnexa. Eye surgery is part of ophthalmology and is performed by an ophthalmologist or eye surgeon. The eye is a fragile organ, and require ...

after a

primus stove The Primus stove was the first pressurized-burner kerosene (paraffin) stove, developed in 1892 by Frans Wilhelm Lindqvist, a factory mechanic in Stockholm. The stove was based on the design of the hand-held blowtorch; Lindqvist's patent covered ...

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 Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...

J. H. C. Whitehead John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princet ...

, and

Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersion (mathematics), immersions, characteristic classes and, ...

) 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 entered

Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...

, where he was strongly influenced by the lectures of

Pavel Alexandrov Pavel Sergeyevich Alexandrov (), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topol ...

who would become his doctoral thesis advisor. After graduating in 1929, he obtained a position at

. In 1934 he joined the

Steklov Institute Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Ste ...

in Moscow. In 1970 he became vice president of the

International Mathematical Union The International Mathematical Union (IMU) is an international organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports the International ...

Work

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 In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

, now called

Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...

. Using these tools, he was able to solve the case of

Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathemat ...

for abelian groups in 1934. In 1935, he was able to compute the homology groups of the classical compact

Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas ...

, which he would later call his greatest achievement. With

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became ...

, he is regarded as one of the co-founders of

cobordism theory In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same d ...

, and co-discoverers of the central idea of this theory, that framed cobordism and stable homotopy are equivalent. This led to the introduction around 1940 of a theory of certain

characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections. Characterist ...

es, now called

Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...

es, designed to vanish on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

that is a boundary. In 1942 he introduced the cohomology operations now called Pontryagin squares. Moreover, in

operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...

there are specific instances of Krein spaces called Pontryagin spaces. Starting in 1952, he worked in

theory. His

maximum principle In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the maximum principle i ...

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'. Pontryagin authored several influential monographs as well as popular textbooks in mathematics. Pontryagin's students include

Dmitri Anosov Dmitri Victorovich Anosov (; November 30, 1936 – August 7, 2014) was a Russian mathematician active during the Soviet Union. He is best known for his contributions to dynamical systems theory. He was a full member of the Russian Academy of Scie ...

, Vladimir Boltyansky, Revaz Gamkrelidze, Yevgeny Mishchenko, Mikhail Postnikov, Vladimir Rokhlin, and Mikhail 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 the

Moscow Mathematical Society The Moscow Mathematical Society (MMS) is a society of Moscow mathematicians aimed at the development of mathematics in Russia. It was created in 1864, and Victor Vassiliev is the current president. History The first meeting of the society w ...

publicly denounced as counter-revolutionary the Society's head

Dmitri Egorov Dmitri Fyodorovich Egorov (; December 22, 1869 – September 10, 1931) was a Russian and Soviet mathematician known for contributions to the areas of differential geometry and mathematical analysis. He was President of the Moscow Mathematical Soc ...

, 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 attacked

Nathan Jacobson Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awa ...

for being a "mediocre scientist" representing the " Zionism movement", while both men were vice-presidents of the

.O'Connor, John J; Edmund F. Robertson
Nathan Jacobson
. MacTutor History of Mathematics archive.
by Lev Pontryagin, Narod Publications, Moscow, 1998 (in

Russian Russian(s) may refer to: *Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *A citizen of Russia *Russian language, the most widely spoken of the Slavic languages *''The Russians'', a b ...

). When a prominent Soviet Jewish mathematician,

Grigory Margulis Grigory Aleksandrovich Margulis (, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic the ...

, was selected by the IMU to receive the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

at the upcoming 1978 ICM, Pontryagin, who was a member of the executive committee of the IMU at the time, vigorously objected.Olli Lehto.
Mathematics without borders: a history of the International Mathematical Union.

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, 1998. ; pp. 205-206 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 in ''

Science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

'' in 1979. In his memoirs Pontryagin claims that he struggled with

Zionism Zionism is an Ethnic nationalism, ethnocultural nationalist movement that emerged in History of Europe#From revolution to imperialism (1789–1914), Europe in the late 19th century that aimed to establish and maintain a national home for the ...

, which he considered a form of

racism Racism is the belief that groups of humans possess different behavioral traits corresponding to inherited attributes and can be divided based on the superiority of one Race (human categorization), race or ethnicity over another. It may also me ...

Publications

* (translated by

Emma Lehmer Emma Markovna Lehmer (''née'' Trotskaia) (November 6, 1906 – May 7, 2007) was an American mathematician known for her work on reciprocity laws in algebraic number theory. She preferred to deal with complex number fields and integers, rather ...

) * 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) ** * 1962 - with Vladimir Boltyansky, Revaz Gamkrelidze, and : The Mathematical Theory of Optimal Processes

Notes

External links