Homotopy theorists have unknowingly been studying ``brave new rings'' since E.H. Brown proved his celebrated Representability Theorem, which in its classical form says that the singular cohomology functors with coefficients in an abelian group are represented by the corresponding family of Eilenberg-Mac Lane spaces-abelian groups ``up to homotopy''. In the first part of the talk, we'll review what all this means and show how it leads directly to the notion of a ring spectrum, which is a direct generalization of the classical notion of (discrete) ring. Along the way, we'll see how category-theory nuts think about algebraic structures like rings in terms of diagrams, monoids, and enrichment.

The study of rings (and, more generally, differential graded algebras) themselves as a special case of ring spectra is called ``brave new algebra''. It is a relatively new part of homotopy theory, having been put on a firm foundational footing only within the last dozen years or so. It provides us with a host of examples of the applications of homotopy-theoretic methods and thinking to other parts of mathematics. In the second part of the talk, we'll do a whirlwind tour of some applications without much in the way of proofs.

I won't assume that you're a devotee of the Way of the Category, but I also won't bother defining topological spaces, rings, chain complexes of modules, categories, functors, and natural transformations. You don't really need to know what cohomology is either, but it will help if you know what de Rham cohomology is.