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From December 1969 to October 1970 he was a visiting scientist at the Lomonosov University in Moscow, and since then, spoke Russian fluently and had many friends from the former Soviet Union.
Krylov studied at Lomonosov University, where he in 1966 under E. B. Dynkin attained a doctoral candidate title (similar to a PhD) and in 1973 a Russian doctoral degree (somewhat more prestigious than a PhD).