On one side is Harvard, the nation’s oldest and wealthiest university, with a brand so powerful that its name is synonymous with prestige.

Confronted with huge cuts to its federal funding, Columbia University’s leaders face a grim decision: They can yield to the Trump administration’s demands over allegations of antisemitism — ceding extraordinary control to the federal government — or they can fight back, potentially risking even more debilitating cuts in an escalating clash.