talk 1: Hopf algebras and periodic localizations of homotopy theory

I will discuss various localizations of unstable homotopy theory arising from the chromatic perspective, namely the v${}_n$-periodic localizations and the localizations with respect to T(n)-homology. There are many ways to approach these through "algebraic models". For example, the v${}_n$-periodic localizations can be studied via the theory of spectral Lie algebras. Quite orthogonally, p-finite spaces can be studied through their cochain algebras with values in Morava E-theory. I will discuss how spectral Hopf algebras provide a framework which subsumes both of these perspectives and describes the interaction between them. We will see some aspects of the structure theory of T(n)-local spectral Hopf algebras, which resembles that of Hopf algebras in characteristic zero.