In 1973, he became the acting director of the University of Wisconsin Army Mathematics Research Center when J. Barkley Rosser retired.
[3] Buck worked on approximation theory, complex analysis, topological algebra, and operations research.
His doctoral students include Lee Rubel and Thomas W. Hawkins, a well-known historian of mathematics.
In 1962 he was an invited speaker (Global solutions of differential equations) at the International Congress of Mathematicians in Stockholm.
Buck was an accomplished amateur pianist and at age 18 won a prize for composition for piano.