In this talk, I will introduce spectra, the basic objects of stable homotopy theory, and explain how they are used to enlarge the category of rings to include objects of interest to algebraic topology.
In ordinary algebra, a ring is an algebra over the integers. In fact, the integers are initial among rings, insofar as any ring receives a unique map from the integers. In stable homotopy theory, the basic objects of study are stable homotopy groups, which are algebraic invariants of a space built from homotopy classes of maps out of spheres. Homotopy theorists then go one step further and define algebras over the sphere spectrum, by way of analogy to rings as algebras over the integers. These ring spectra, as they are called, are ubiquitous in modern homotopy theory.