Krutkov worked on cosmology, quantum theory, statistical mechanics, tensor fields, and other areas.
Krutkov was born in St. Petersburg where his father Alexander taught Greek, Latin and Russian.
Funded by a Rockefeller Foundation grant, he worked in Germany and Holland in 1922-23 during which time he met Albert Einstein.
Here he pointed out an error in Einstein's critique of Friedmann's work on the expanding universe.
During World War II he was imprisoned but continued to work on some aerodynamics problems with A.N.