Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic K-theory.  In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of Bökstedt-Hsiang-Madsen to the infinity-category theoretic methods of Nikolaus-Scholze.

Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic K-theory.  In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of Bökstedt-Hsiang-Madsen to the infinity-category theoretic methods of Nikolaus-Scholze.

I will define operads and modules over operads, in the context of both chain complexes and topological spaces. I will then describe a number of important examples of these structures arising in algebraic topology and explain their significance and utility.