Working session: Topological cyclic homology. Abstracts from the working session held April 1--7, 2018 (Q2420936)

Summary: Introduced by Bökstedt-Hsiang-Madsen in the nineties, topological cyclic homology is a manifestation of the dual visions of Connes-Tsygan and Waldhausen to extend de Rham cohomology to a noncommutative setting and to replace algebra by higher algebra. The cohomology theory that ensues receives a denominator-free Chern character from algebraic \(K\)-theory, used by Hesselholt-Madsen to evaluate the \(p\)-adic \(K\)-theory of \(p\)-adic fields. More recently, Bhatt-Morrow-Scholze have defined a ``motivic'' filtration of topological cyclic homology and its variants, the filtration quotients of which give rise to their denominator-free \(p\)-adic Hodge theory \(\mathbb{A}\Omega\).