Lev Pontryagin

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. 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.[1] 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.

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 the applied control at each moment is either the maximum 'steer', or none.

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

Monument to Lev Pontryagin on wall of building on Leninsky Prospekt in Moscow, where he lived from 1939 to 1988.

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

Pontryagin was accused of anti-Semitism on several occasions. 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.[2][3] He rejected charges of anti-Semitism in an article published in Science in 1979,[4] claiming that he struggled with Zionism, which he considered a form of racism.[3] 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.[5] 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.[5] Pontryagin also participated in a few notorious political campaigns in the Soviet Union, most notably, in the Luzin affair.

• Pontrjagin, L. (1939), Topological Groups, Princeton Mathematical Series, 2, Princeton: Princeton University Press, MR 0000265
• 1962 - Ordinary Differential Equations
• 1962 - The Mathematical Theory of Optimal Processes
• 1963 - Foundations of Combinatorial Topology

• Andronov–Pontryagin criterion
• Kuratowski's theorem, also called the Pontryagin–Kuratowski theorem
• Pontryagin class
• Pontryagin duality
• Pontryagin's maximum principle

• Lev Pontryagin at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Lev Pontryagin", MacTutor History of Mathematics archive, University of St Andrews.
• Autobiography of Pontryagin (in Russian)