Sean Tilson: Power operations and Commutative ring spectra

We will begin by reviewing and constructing power operations in the familiar setting of chain complexes. In stable homotopy, these operations help distinguish different geometric objects. These operations are also the residue of a rich homotopical structure. We will also define such structure and explain its role in stable homotopy theory. This first talk will be accessible to graduate students and those familiar with derived categories. Such power operations also act on the homotopy of highly structured ring spectra. We will compute these operations on relative smash products using the Kunneth spectral sequence. We will interpret the homotopy of these relative smash products and the algebra of operations in terms of different realizations of highly structured DGAs. We will also discus the relation to the relevant notion of cotangent complexes. This second talk will be geared towards an algebro-geometric audience.