In 1958 Brown earned his Ph.D. from the University of Wisconsin-Madison under R. H. Bing. From 1960 to 1962 he was at the Institute for Advanced Study. Afterwards he became a professor at the University of Michigan at Ann Arbor.
With Barry Mazur in 1965 he won the Oswald Veblen prize for their independent and nearly simultaneous proofs of the generalized Schoenflies hypothesis in geometric topology. Brown's short proof was elementary and fully general. Mazur's proof was also elementary, but it used a special assumption which was removed via later work of Morse.
- Brown, Morton (1960). "A proof of the generalized Schoenflies theorem". Bull. Amer. Math. Soc. 66 (2): 74–76. doi:10.1090/s0002-9904-1960-10400-4. MR0117695
- List of Fellows of the American Mathematical Society, retrieved 2012-11-10.