President Obama had just disembarked from Air Force One and was still on the tarmac in Rome when he turned to his host, John R. Phillips, the American ambassador to Italy, with an unexpected request: How about a dinner party tomorrow night? Over the next 24 hours, the startled Mr. Phillips and his wife, the former Obama aide Linda Douglass, scrambled to gather some of Italy’s intellettuali. The architect Renzo Piano flew in from Genoa. The particle physicist Fabiola Gianotti arrived from Geneva. John Elkann, the chairman of Fiat and the owner of the Italian soccer club Juventus, came too,...