Title: Localization and algebraic structure in homotopical algebra.
Speaker: David White (Wesleyan University)
Abstract: Modern homotopy theory is a blend of algebra and topology, and in recent years a number of results have demonstrated the value of doing homotopy theory in a monoidal setting so that algebraic techniques may be used. Perhaps the most striking example is the Hill-Hopkins-Ravenel resolution of the Kervaire Invariant One Problem, which made use of the model category of equivariant spectra and demonstrated the computational strength of equivariant commutative ring spectra. A key step in the proof relied on the commutativity of a particular localization of a commutative ring spectrum, but a recent example due to Mike Hill demonstrates that general localizations need not preserve commutative structure.

In this talk we will give a broad overview of the concept of localization in algebra, category theory, and topology, focusing on a treatment via universal properties. We will motivate the definition of a model category and of Bousfield localization. We will then give some new results regarding the interplay of Bousfield localization with monoidal structure, with a particular focus on when localization preserves algebraic structure. We will end with a characterization of when commutativity is preserved in the case of equivariant spectra, resolving Hill's example in the process.