Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a

Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...

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, structure, space, models, and change. History On ...

. He was born in

Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ...

and lost his eyesight completely due to an unsuccessful eye surgery 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 cove ...

explosion when he was 14. Despite his blindness he was able to become one of the greatest mathematicians of the 20th century, partially with the help of his mother Tatyana Andreevna who 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 topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...

, J. H. C. Whitehead, 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, immersions, characteristic classes, and geometric integration t ...

) to him. He made major discoveries in a number of fields of mathematics, including

optimal control Optimal control theory is a branch of mathematical optimization 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 ...

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 ...

and

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 ...

Work

Pontryagin worked on duality theory for

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

while still a student. He went on to lay foundations for the abstract theory of the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

, now called

Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), 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 numb ...

. 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 w ...

, 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 ...

, 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 the bundle is "twisted" and whether it possesses sections. Characteristic classes ...

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 Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

. In 1942 he introduced the cohomology operations now called

Pontryagin square In mathematics, a Pontryagin cohomology operation is a cohomology operation taking Characteristic class, cohomology classes in ''H''2''n''(''X'',Z/''p'r''Z) to ''H''2''pn''(''X'',Z/''p'r''+1Z) for some prime number ''p''. When ''p''=2 these op ...

s. 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 operators ...

there are specific instances of

Krein space In mathematics, in the field of functional analysis, an indefinite inner product space :(K, \langle \cdot,\,\cdot \rangle, J) is an infinite-dimensional complex vector space K equipped with both an indefinite inner product :\langle \cdot,\,\cdo ...

s called

Pontryagin space In mathematics, in the field of functional analysis, an indefinite inner product space :(K, \langle \cdot,\,\cdot \rangle, J) is an infinite-dimensional complex vector space K equipped with both an definite bilinear form, indefinite inner product ...

s. Later in his career he worked in

theory. His

maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

is fundamental to the modern theory of optimization. He also introduced there 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 (russian: Дми́трий Ви́кторович Ано́сов; November 30, 1936 – August 7, 2014) was a Russian mathematician active during the Soviet Union, he is best known for his contributions to dynamical syste ...

Vladimir Boltyansky Vladimir Grigorevich Boltyansky (russian: Влади́мир Григо́рьевич Болтя́нский; 26 April 1925 – 16 April 2019), also transliterated as Boltyanski, Boltyanskii, or Boltjansky, was a Soviet and Russian mathematician, ...

Revaz Gamkrelidze Revaz Valerianovic Gamkrelidze ( ka, რევაზ ვალერის ძე გამყრელიძე, ISO 9984: ''Revaz Valeris je Gamqrelije''; born February 4, 1927) is a Georgian and Soviet mathematician known for his work in opt ...

, Evgeni Mishchenko, Mikhail Postnikov, Vladimir Rokhlin, and Mikhail Zelikin.

Controversy and anti-semitism allegations

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 awar ...

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

International Mathematical Union The International Mathematical Union (IMU) is an international non-governmental 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 ...

.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) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...

). He rejected charges of anti-Semitism in an article published in ''

Science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...

'' in 1979, claiming that he struggled with

Zionism Zionism ( he, צִיּוֹנוּת ''Tsiyyonut'' after ''Zion'') is a Nationalism, nationalist movement that espouses the establishment of, and support for a homeland for the Jewish people centered in the area roughly corresponding to what is ...

, 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 over another. It may also mean prejudice, discrimination, or antagonism ...

. When a prominent Soviet Jewish mathematician,

Grigory Margulis Grigory Aleksandrovich Margulis (russian: Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on ...

, was selected by the

IMU IMU may refer to: Science and technology * Inertial measurement unit, a device that measures acceleration and rotation, used for example to maneuver modern vehicles including motorcycles, missiles, air- and spacecraft Businesses and organizations ...

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 the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

at the upcoming 1978 ICM, Pontryagin, who was a member of the Executive Committee of the

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 also participated in a few notorious political campaigns in the Soviet Union, most notably, in the Luzin affair.

Publications

* (translated by

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

) * 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

, and : The Mathematical Theory of Optimal Processes

Notes

External links

* *
Autobiography of Pontryagin
(in Russian)

* ttp://www.math.nsc.ru/LBRT/g2/english/ssk/case_e.html Kutateladze S. S., The Tragedy of Mathematics in Russia {{DEFAULTSORT:Pontryagin, Lev Semenovich 1908 births 1988 deaths Blind people from Russia Control theorists Heroes of Socialist Labour Soviet mathematicians Full Members of the USSR Academy of Sciences Blind academics 20th-century Russian mathematicians Soviet people Moscow State University faculty

Work

Controversy and anti-semitism allegations

Publications

See also

Notes

External links