His research focused on dynamical systems, the topology of curve families, complex function theory, and differential equations.

[3] Kaplan received a Rogers Fellow scholarship to study in Europe from 1936-1937, during his second year of graduate school.

[3] Upon returning to the United States, Kaplan accepted a yearlong teaching fellowship at Rice Institute for the 1938-1939 school year, thus completing his graduate program.

[4] While attending lectures at the Eidgenössische Technische Hochschule (ETH) Zürich he met a fellow mathematician, Ida Roetting, whom he nicknamed Heidi and would eventually marry in 1938.

In June 1944, Kaplan worked at Brown University as a researcher in an Applied Mathematics Group for the Taylor Model Basin, the Watertown Arsenal and the Bureau of Ordnance of the Navy Department.

Kaplan taught a variety of undergraduate and graduate courses during his time at Michigan, and further advised eight doctoral students.

He was a superb expositor, and his ability to elegantly convey mathematical ideas explains the enormous impact of his textbooks.

He urged other textbook authors to use clear and simple language whenever possible, in order to “make the more advanced material accessible to those with limited background.

[3]” He also taught a class called "Mathematical Ideas in Science and the Humanities," which focused on the use of math as an instrument to organize thinking about complex problems.

In the proposal he outlined a variety of topics to be explored in the study: the historical background of higher education including tuition rates, enrollment rates, and changes in social customs; a study of changing demographics; the economic need for college-trained people for the betterment of society; and a thorough account of the present resources available to higher education and how these could be modified for greater efficiency.

In the 1990s, Kaplan's correspondence and reports focused heavily on the "grave difficulties" between faculty government leaders (members of the Senate Assembly) and the top administration offices, specifically President Duderstadt and Provost Whitaker.

While serving on the Senate Advisory Committee on University Affairs, Kaplan received their Distinguished Faculty Governance Award in 1986.

In the years to come he would receive several requests from the University Regents Commission to return to active duty to teach specific classes in the mathematics department.

[2][6] The film Keeping in Mind, an account of the mistreatment of the three professors who were suspended for their unpopular views during the McCarthy era, was played in the spring of 1989.

After an audience member suggested the University make amends for its mistreatment of the three professors, the AAUP pursued this goal.

[6] After his wife's death in 2005, Kaplan wrote a book titled Bill and Heidi: Beginning of our Lives Together, which was a translated composition of all their early correspondence before their wedding.

Often, he would quietly remind his associates when they had strayed from their stated purpose or point out a legal or historical obstacle to what was being considered.