J. Peter May is a homotopy theorist at the University of Chicago, inventor of operads as a technique for studying infinite loop spaces and spectra. He is now one of the main advocates of higher category theory in the United States.
Peter May’s webpage.
Peter May: Wikipedia entry.
stable homotopy theory
equivariant homotopy theory (Bredon cohomology, equivariant stable homotopy theory, rational equivariant stable homotopy theory)
higher category theory