Brave New Rings

The notion of a ring, or number system, is fundamental to mathematics. Modern algebraic topology has generalized the classical notion of a ring as it occurs in number theory or algebraic geometry, to encompass a larger class of "brave new rings" technical ly known as S-algebras. The generalization is justified by applications to geometric topology. These new rings also have a form of arithmetic and geometric structure, expressed in the language of stable homotopy theory and algebraic K-theory, for which new properties and phenomena have recently been uncovered. For example, algebraic K-theory may induce a so-called chromatic red-shift of the periodic patterns that organize stable homotopy theory, and may descend gracefully along certain covering spaces known as Galois extensions of S-algebras. The project will bring together a research team of post docs and senior researchers to explore a program for resolving these mathematical questions and conjectures, as posed by the project leader.