On mathematical creative work of L.S. Pontryagin (By D.V. Anosov)




Lev Semenovich Pontryagin was born in Moscow on September, 3, 1908. At the age of 14 he lost his sight in an accident. This did not prevent him from becoming one of the greatest mathematicians of the XX century.

L.S.Pontryagin, P.S.Alexandrov and A.N.Kolmogorov
Photo by A.I.Pontryagina

What follows is a short review on some of L.S.'s achievements presented in his key paper series.
L.S. started his first paper series as he was a student. His work was devoted to duality in algebraic topology. At that time, the duality of PoincareVeblen for the manifold M^{n} and the duality of Alexander for a polyhedron in R^{n} (or in a sphere) were already known; one represented them in the form of equalities between numerical invariances. L.S. gave a modern interpretation and expressed the dualities in terms of the groups of homologies (he also treated the Alexander duality for a polyhedron in M^{n}). Next, he passed from a polyhedron to a compact set (first, in R^{n}, and then in M^{n}) and obtained an ultimate result through the introduction of homologies with compact groups of coefficients. These papers by L.S. had not only the immediate significance. They (and the papers of H.Hopf on continuous mappings, which appeared in approximately the same period) clearly demonstrated the advantages of the technique of the groups of homologies (the idea of introducing these groups was just put forward by E.Noether) in comparison with the associated numerical invariances technique. Finally, these papers by L.S. showed that sometimes an isomorphism between the homologies is determined uniquely due to its "natural" character, and sometimes (for example, if the homologies are finite dual groups) it exists due to other reasons and is by no means determined uniquely. The origin of these facts, which are common knowledge nowadays, has been nearly forgotten.

L.S.Pontryagin at his typewriter

Addressing the issue of compact groups of coefficients stimulated L.S. to start investigations on topological algebra; he is the actual founder of this discipline. (However, his first paper on this subject, which gives a topological characterization of the fields R and C and the body of quaternions, L.S. wrote independently.) His principal results in this field deal with commutative compact and locally compact groups; L.S. explored their structure and worked out the associated harmonic analysis involving the "Pontryagin's duality" between a group and the group of its characters ( L.S. studied duality between compact and discrete groups. An extension to locally compact groups was given by Van Kampen ).
The next important series of papers by L.S. is devoted to homotopic and  in modern terms  differential topology. He discovered a relationship between the homotopy problems and the problems on smooth manifolds, and found new invariances of the smooth manifolds, which are nowadays known as characteristic classes of Pontryagin. A fundamental significance of these achievements leaves no space for comments. One should only note that in this paper series (as well as in contemporary papers by H.Hopf, E.Stiefel, H.Witney, and somewhat later papers by S.Eilenberg, N.Steenrod and Sh.Chern) an essential part of the theory of skew products was created, and cohomology operations appeared, which were not of a usual algebraic type.
In a period between these two series of works, L.S. defined groups of homologies of compact classical Lie groups using the Morse theory (which was, probably, a first implementation of the Morse theory in such context).

E.F.Mishchenko, L.S.Pontryagin and S.M.Nikolskii
Photo by A.I.Pontryagina

In the beginning of the 50s, L.S. went into (broadly understood) theory of ordinary differential equations, which had episodically drawn his interest earlier. (The "episodic" studies of L.S. were by no means "episodes" for the theory: one of them resulted in a  joint with A.A.Andronov  paper on structurally stable systems!) L.S. started systematic studies in this field; several series of papers appeared.
The first paper series was dedicated to singular perturbations, namely, the relaxation oscillation systems with a small parameter by the derivative.
The second one had most serious consequences. In this series of works, a mathematical theory of optimal processes was created. The role of the "maximum principle" discovered by L.S. (and rigorously proved by V.G.Boltyanskii) is clearly seen in the context of the classical variational problems, for which the "maximum principle" comprises the EulerLagrange equations, the Weierstrass condition and the angle conditions of WeierstrassErdmann. In contrast to these classical conditions, the "maximum principle" is applicable to a number of nonclassical problems.
The third series of papers was devoted to differential games. R.Issaacs was to a certain extend a predecessor of L.S. in this field. However, only due to L.S. theory of differential games was shaped into a powerful scientific stream.
L.S. performed alone a majority of his investigations on topology and related issues. In his "differential" period, his principal coauthors were E.F.Mishchenko (singular perturbations), V.G.Boltyanskii and R.V.Gamkrelidze (mathematical theory of optimal processes), and E.F.Mishchenko and A.S.Mishchenko (differential games).

Yu.V.Prokhorov, V.A.Melnikov, S.P.Novikov,
S.M.Nikolskii, L.S.Pontryagin, A.N.Tikhonov, G.I.Marchuk
September, 3, 1983
Pontryagin's country house near Moscow
Photo by A.I.Pontryagina

Lev Semenovich Pontryagin died on May, 3, 1988. He was buried at the Novodevichie Memorial Cemetery in Moscow.

Translated by A.V. Kryazhimskii 





