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. Using these tools, he was able to solve the case of
Hilbert's fifth problem for abelian groups in 1934. In 1935, he was able to compute the homology groups of the classical compact
Lie groups, which he would later call his greatest achievement. 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. 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. Starting in 1952, he worked in
optimal control theory. His
maximum principle 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,
Vladimir Boltyansky,
Revaz Gamkrelidze, Yevgeny Mishchenko,
Mikhail Postnikov,
Vladimir Rokhlin, and
Mikhail Zelikin. ==Controversy and antisemitism allegations==