He remained there, doing research on asymptotic methods and singular perturbations with Joseph Keller and a number of other stimulating colleagues and students.

O'Malley spent a year at the University of Edinburgh, where his lecture notes formed the basis of his book, Introduction to Singular Perturbations (Academic Press, 1974).

[3] O'Malley's research emphasized the relationship between singular perturbation theory and various regularization methods for differential-algebraic systems, geometric approaches to understanding the limiting solutions to singularly perturbed boundary value problems, the motion of shock layers and other interfaces, the interplay between asymptotic and numerical methods, and tough problems of asymptotic matching.

Toward the end of his life, he continued to collaborate with an international collection of interesting characters, and received support for his scholarly work from the National Science Foundation.

Among his other positions with SIAM he had been program chair for several meetings including ICIAM `91, the second international industrial and applied mathematics conference, which drew 2,200 participants.