His research included functional analysis, partial differential equations and mathematical physics.
Birman was born in 1928 in Leningrad (now St. Petersburg) to a father that was a professor and specialist in theoretical mechanics and a mother that was a school teacher.
Although Birman was one of the best students, he was denied a doctorate at Leningrad University because of his Jewish origins and the state antisemitic policy of the time.
In 1974 he was an invited speaker at the International Congress of Mathematicians in Vancouver, but was prohibited to leave the Soviet Union.
Birman was the supervisor for more than 20 students, including Boris Pavlov and Timo Weidl.