Shnirelman was appointed to the chair of mathematics at the Don Polytechnic Institute in Novocherkassk in 1929. Then, in 1930, he returned to Moscow University but went to study at Göttingen in 1931. The paper  describes his stay there based on two of his letters he wrote to his mother which are reproduced in . After the Göttingen visit, he returned to Moscow to continue teaching at the University but, after election to the Soviet Academy of Sciences in 1933, he worked at the Mathematical Institute of the Academy beginning in the following year. He was only 33 years old when he died in 1938, having made deep contributions to two quite separate areas of mathematics.
There is some doubt about how he died. According to Pontryagin's autobiography  L A Lyusternik reported that Shnirelman gassed himself, probably because he was depressed that he could no longer prove results as good as the ones he had found earlier in his career. However, according to the E B Dynkin, in an interview he gave in 1988  the mathematician and historian Sofia Alexandrovna Yanovska reported that Schnirelman had been recruited by the NKVD (the Soviet intelligence service) and was shot by them.
L A Lyusternik became a friend and important collaborator with Shnirelman and together they made significant contributions to topological methods in the calculus of variations in a series of paper written jointly between 1927 and 1929. These include the three joint papers published in 1929: Sur un principe topologique en analyse; Existence de trois géodésiques fermée sur toute surface de genre 0 Ⓣ; and Sur le problème de trois géodésiques fermée sur les surfaces de genre 0 Ⓣ. Also an important contribution to this topic is their joint paper Topological methods in variational problems (1930). Youschkevitch writes :-
Their starting point was Poincaré's problem of the three geodesics, which they first solved completely and generally by showing the existence of three closed geodesics on every simply connected surface (every surface homeomorphic to a sphere). For the proof of this theorem the authors used a method, which they broadly generalised, that had been devised by G D Birkhoff, who in 1919 showed the existence of one closed geodesic. Shnirelman and Lyusternik also applied their "principle of the stationary point" to other problems of geometry "im Grossen". They also presented a new topological invariant, the category of point sets.In 1930 Shnirelman introduced important new ideas into number theory. Using these ideas of compactness of a sequence of natural numbers he was able to prove a weak form of the Goldbach conjecture showing that every number can be written as a sum of at most C primes where C is a definite number. He was able to show that C < 300 000. Later mathematicians have been able to find much better bounds for C. He presented these ideas in Über additive Eigenschaften von Zahlen Ⓣ published in Mathematische Annalen in 1933. Concerning this paper Halberstam wrotes over 50 years later:-
... Shnirelman had many more results and ideas than those that were eventually to find their way into the standard texts, and ... mathematicians might well find a visit to his classical memoir rewarding.In a talk delivered at a meeting of the German Mathematical Society on the morning of 17 September 1931, Shnirelman first reported on his now famous researches in additive number theory which were to be presented in the 1933 paper given above. His talk, given in German, appears in Russian translation in . The Goldbach conjecture that every number is the sum of at most 3 primes still appears to be open. Later significant contributions by Shnirelman include his two papers On the additive properties of numbers, and On addition of sequences published in 1940 after his death.
I M Gelfand, in an interview on 11 January 1992 at the Joint Mathematics Meetings in Baltimore, was asked " Who was most influential on you as a young man?" Gelfand replied "Many people, but especially Shnirelman: A person must learn from everyone in order to be original."
Article by: J J O'Connor and E F Robertson