Keller worked on the application of mathematics to problems in science and engineering, such as wave propagation. He contributed to the Einstein–Brillouin–Keller method for computing eigenvalues in quantum mechanical systems.
Keller was awarded a Lester R. Ford Award (shared with David W. McLaughlin) in 1976 and (not shared) in 1977. In 1988 he was awarded the U.S. National Medal of Science, and in 1997 he was awarded the Wolf Prize by the Israel-based Wolf Foundation. In 1996, he was awarded the Nemmers Prize in Mathematics. In 1999 he was awarded the Ig Nobel Prize for calculating how to make a teapot spout that does not drip. With Patrick B. Warren, Robin C. Ball and Raymond E. Goldstein, Keller was awarded an Ig Nobel Prize in 2012 for calculating the forces that shape and move ponytail hair. In 2012 he became a fellow of the American Mathematical Society.