Krylov was born in
Ustyuzhna,
Vologda Governorate, of the
Russian Empire. He graduated in physics from the
Leningrad University. He then was a
doctoral student in the
Leningrad University's
theoretical physics group supervised by
Vladimir Fock, and wrote thesis on the foundations of
statistical mechanics entitled
Mixing processes in phase space awarded by for the degree of
Candidate of Science in 1941. During the
German invasion of the Soviet Union and the
Siege of Leningrad, Krylov was assigned to the air defense of the city. He continued research work at
Kazan for the
Physical-Technical Institute of the
Academy of Sciences of the Soviet Union, when the Institute was relocated due to the siege, while on active duty and defended dissertation
The processes of relaxation of statistical systems and the criterion of mechanical instability awarded by the degree of
Doctor of Science the following year. He then worked at the various Soviet Union's academic institutes, in 1947 together with his supervisor coauthored the
Fock-Krylov theorem on quasi-stationary state
decay in
quantum mechanics, returned to Leningrad, but fell ill in 1946 and died due to
sepsis caused by a
streptococcus. which contained Fock's biographical note, description of Krylov's theoretical views by both Migdal and Fock. Wightman's preface compared Krylov's ideas to the results by
Poincaré,
Birkhoff,
von Neumann,
Hopf,
Kolmogorov,
Khinchin, and
Wiener, whereas the book was compared by him to the classical books in statistical mechanics by
Maxwell,
Boltzmann,
Gibbs, and
Ehrenfest and
Afanasjewa-Ehrenfest. Thanks to this effort, Krylov's research results had received a permanent place in modern theoretical physics and have laid the foundations of
dynamical systems theory and
quantum mechanics. == Selected publications ==