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, 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 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 either the maximum 'steer' should be applied to a system, or none.
Pontryagin authored several influential monographs as well as popular textbooks in mathematics. The fact that Pontraygin was legally blind since the age of 14, further underscores his scientific achievements.
Pontryagin's students include Dmitri Anosov, Vladimir Boltyansky, Evgeni Mishchenko, Mikhail Postnikov and Vladimir Rokhlin.
Read more about this topic: Lev Pontryagin
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