Other businesses, including Nvidia and Intel, have been drawn into making financial deals with the Trump administration in order to continue doing business or to sell their products to China. The specific details of most of those deals have not been made public.

2. Redefining discrimination

On his first day in office, Mr. Trump signed executive orders gutting racial equity policies and protections for transgender people. Those themes, along with addressing antisemitism and targeting international students, were evident in many of these agreements.

No “unlawful D.E.I. goals”: Columbia and Brown agreed to eliminate programs that aim to achieve diversity goals. (Both universities had already eliminated race-conscious affirmative action following a June 2023 Supreme Court decision outlawing it.) They promised to rely more on quantitative measures, instead of demographics, in their admissions practices.

Experts say relying on test scores and grades in admissions could result in wealthier, less diverse student populations at these elite institutions.

Law firms were similarly subject to these rules in their hiring practices.

Transgender students: The University of Pennsylvania has been central in the debate around transgender athletes, specifically because of Lia Thomas, a transgender swimmer who graduated in 2022 and held several of Penn’s swimming records. The school’s deal with Mr. Trump revoked her records and limited how transgender students may participate in its athletic programs.

In this way, the university bowed to the administration’s new interpretation of Title IX, a law that until recently protected transgender athletes from sex discrimination in education.

Penn and the other universities also agreed to additional rules around single-sex facilities and medical services for transgender students.

According to Brown, the number of minors enrolled at the university is typically less than 10 percent of all first-year undergraduates; the campus does not have surgical facilities; and its doctors do not typically prescribe puberty blockers.

Fighting antisemitism: Columbia agreed to pay $21 million to a compensation fund to resolve alleged civil rights violations against its Jewish employees. It also agreed to a review of its regional studies programs, starting with the Middle East, to ensure that they are “comprehensive and balanced.” The agreement does not define how those terms will be applied.

The school will also appoint new faculty members who will have joint positions in both the Institute for Israel and Jewish Studies and other departments, and add a student liaison to coordinate and advise on antisemitism issues.

Both Columbia’s and Brown’s agreements have provisions outlining support for Jewish life on campus.

International students: Columbia’s agreement with the Trump administration outlines provisions on international students, including asking them their reasons for wanting to study in the United States and reducing the school’s reliance on international student enrollment. (Columbia has about 13,700 international students, about 38 percent of its total student body.)

3. Government oversight

Through the deals with Columbia and Brown, the Trump administration also gained access to information about their applicants, including details on race, grades and test scores.

Both schools are also required to make anonymized information on enrolled students available to the public, including demographics and grade point averages of each class.

Columbia also agreed to pay for a monitor, approved by the school and the government, to ensure that both sides abide by their commitments. It must now also inform the Department of Homeland Security when an international student is arrested. (Universities were already required to inform Homeland Security when an international student was suspended or expelled.)

Brown agreed to hire an external organization to conduct a campus survey by the end of the year on the school’s climate for Jewish students.

Similarly, the law firms also agreed to hire outside counsel to ensure adherence.

4. Assessing ideology

Some of the law firms targeted by Mr. Trump were associated with lawyers who have previously investigated him, or who have worked closely with those who did. Other firms had prominent Democrats on staff, or employed people who frequently criticized the president.

To that end, in each of their agreements, the law firms agreed to work on a wider range of cases, regardless of the political affiliation of the lawyer or prospective client.

In return for the concessions, Mr. Trump revoked his executive order against the law firm Paul Weiss that would have suspended its security clearances, restricted its access to federal buildings and threatened its contracts with the government. The eight other law firms struck deals pre-emptively to avoid being subject to similar executive orders.

For most of the universities, the administration restored hundreds of millions of dollars in federal funding that it had previously frozen. It also closed pending investigations into the schools of antisemitism or alleged violations of civil rights. Both the presidents of Columbia and Brown have publicly stated that these deals preserve the schools’ academic freedom.

What’s next

Like Harvard, four major law firms have fought back against the president instead of striking a deal. Perkins Coie, WilmerHale, Susman Godfrey, and Jenner & Block all filed lawsuits, which resulted in federal judges temporarily blocking Mr. Trump’s executive orders targeting them. The administration has recently begun to appeal these rulings.

Several other law firms have been questioned by the U.S. Equal Employment Opportunity Commission on their hiring practices and potential discrimination against white candidates, raising concerns of threats from the White House.

The Trump administration has vowed to appeal the court’s ruling that its funding cuts against Harvard were illegal. It remains unclear whether the school will receive its money. Mr. Trump has also frozen federal funding to other universities, including Princeton, Cornell, Duke, Northwestern and the University of California, Los Angeles, setting the stage for potential negotiations. In August, the administration proposed that U.C.L.A. pay more than $1 billion to reach a settlement. This month, after a difficult tenure that included attacks from Republicans in Congress and funding cuts, the president of Northwestern resigned.

Several other schools are watching for funding cuts as they come under the scrutiny of the Department of Education and a government task force that says it is devoted to rooting out antisemitism.