John Lind: Diagram spaces and the descending central series of the sphere spectrum

Diagram spaces are functors from small combinatorial or geometric categories to the category of spaces.  For appropriate diagram categories, diagram spaces provide a symmetric monoidal model category in which the monoids (resp. commutative monoids) model A-infinity (resp. E-infinity) spaces.  I will summarize work of myself and others in setting up the general theory, and motivate with the basic examples.  As an application, I will then discuss an interesting filtration of BU and BO related to the descending central series of free groups and show how to use diagram spaces to enhance it to a filtration by commutative ring spectra of the corresponding K-theory spectra KU and KO.  In fact, the same techniques give a filtration by commutative ring spectra of any algebraic K-theory spectrum, including the sphere spectrum.