In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle.
 It was inspired by, but is distinct from, the Hamiltonian of classical mechanics.
 Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian.
 For details see Pontryagin's maximum principle.
Lev Semyonovich Pontryagin (Russian: Лев Семёнович Понтрягин , also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician.
 He was born in Moscow and lost his eyesight due to a primus stove 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, J. H. C. Whitehead, and Hassler Whitney) to him.
 He made major discoveries in a number of fields of mathematics, including algebraic topology and differential topology.
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations.
 It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative.
 The theory is named for two men, the Soviet mathematician Mark Grigorievich Krein, and the Japanese Tadao Tannaka.
 In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category Π("G") with some additional structures, formed by the finite-dimensional representations of "G".
In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin.
 It applies to the homology theory properties of the complement of a subspace "X" in Euclidean space, a sphere, or other manifold.
 It is generalized by Spanier-Whitehead duality.
In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes.
 The Pontryagin class lies in cohomology groups with degree a multiple of four.
 It applies to real vector bundles.
In probability theory, the Markov–Krein theorem gives the best upper and lower bounds on the expected values of certain functions of a random variable where only the first moments of the random variable are known.
 The result is named after Andrey Markov and Mark Krein.
Pontryagin's maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.
 It was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students.
 It has as a special case the Euler–Lagrange equation of the calculus of variations.
The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane.
 It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
Mark Grigorievich Krein (Ukrainian: Марко Григорович Крейн , Russian: Марк Григо́рьевич Крейн ; 3 April 1907 – 17 October 1989) was a Soviet Jewish mathematician, one of the major figures of the Soviet school of functional analysis.
 He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory.

Question: Were Mark Krein and Lev Pontryagin from the same country?
Answer: yes