I have been reading this blog post which mentions that:
during World War II Kolmogorov applied his mathematical gifts to artillery problems, helping to protect Moscow from German bombardment.
So, what were the mathematical solutions did Kolmogorov use in helping protect Moscow in the World War 2?
I think, the reference is to Kolmogorov's work on anti-aircraft artillery in 1942 (the theory of artificial scattering of shells).
The following is taken from the article
V.I. Arnolʹd, A. N. Kolmogorov and the natural sciences. Uspekhi Mat. Nauk 59 (2004), no. 1(355), 25–44; translation in Russian Math. Surveys 59 (2004), no. 1, 27–46.
Pages 35-36 (of the English trianslation):
During a state examination on military training, a general of artillery, who came to Moscow State University to evaluate the graduates, asked me: “Who is the best mathematician in Russia at present?” I mentioned Kolmogorov, and the general was pleased: “He did a lot of useful things for us as well, we remember it and also appreciate him.”
Kolmogorov told me with pleasure that in 1942 the Artillery Command requested that he come from Kazan (to which the Academy of Sciences had been evacuated) to Moscow for consultations; according to him, Kolmogorov got a sofa as a lodging (as he told me) in Neskuchnyi garden (in the building of the Presidium of the Academy of Sciences) and worked out a statistical basis for his suggestions about how to organize antiaircraft fire against the massive bombing raids of the enemy. These suggestions, called a theory of artificial scattering of shells, show that under certain conditions it is preferable to shoot at random rather than aim, creating a curtain of shell bursts in the path of the enemy aircraft. The point is that, when trying to hit an individual aircraft, several antiaircraft guns or even a whole battery of guns can choose the same target, and then the nearby enemy aircraft would remain undamaged (if one tries to hit only chosen targets).
A caveat: Many things claimed by Arnold should be taken with a grain of salt. But this story, I think, is solid.
The Wikipedia article on Kolmogorov is a good starting point and will give you some search terms that could be fruitful.
For example, Shiryaev A.N. (2003) On the Defense Work of A. N. Kolmogorov during World War II. In: Booß-Bavnbek B., Høyrup J. (eds) Mathematics and War. Birkhäuser, Basel.
This survey references his 1941/42 paper "Estimation of the center and spread of dispersion for a bounded sample"
In the end of 1941 and beginning of 1942, the Soviets had multiple problems with artillery.
The first one was shell and batteries availability: there was not enough of them for field artillery. It was not that much a problem since the mud, and later the snow, reduced the capacity of artillery to inflict damages. So the Soviets relied on infiltration tactics with mobile units made for low intensity fights: light armoured cars, cavalry units, skiing units, airborne troops...
The second problem they faced was numerous air attacks made by the Luftwaffe againt the cities. Despite primitive radars and an increasing number of airplanes, front line operations were the priority and Soviets had to rely a lot on anti aircraft artillery to repel German raids. This artillery needed multiple data to be effective: how airplane behave, deflection, ballistic, how to stop a group of planes rather than destroy each airplane one by one (which is impossible in fact).... According to colonel Proektor, out of 4 000 airplanes targeting Moscow during eponym battle, only 120 airplanes actually reached and bombed targets over Moscow.
So I don't have primary sources on Kolmogorov, but it is probable that he helped in anti aircraft artillery based in Moscow rather than field artillery.
