A cohomology theory E is particularly useful when we can understand its cocycles E^*(X) in terms of geometric objects associated to the space X. A basic example is the description of topological K-theory in terms of complex vector bundles. I will give an analogous interpretation of cocycles for E=K(R), the algebraic K-theory of an associative ring spectrum, in terms of bundles of R-modules over X. This project is motivated by a desire to understand elliptic cohomology theories and higher categorical geometry. I hope to give a gentle introduction to some of these ideas along the way.

For more information, contact Khalid Bou-Rabee khalidb[at]umich(dot)edu