Topological Hochschild homology(THH), which is a lift of Hochschild homology to the stable category, has recently seen a large paradigm shift in the way it and related invariants are studied.
This is due to several groundbreaking results, among which are the results of Bhatt, Morrow, and Scholze which introduce the quasisyntomic topology and compute the quasisyntomic filtration of topological Hochschild homology, topological negative cyclic homology ($TC^{-}$), topological periodic homology(TP), and topological cyclic homology(TC). Building on these results, I will talk about the quasisyntomic filtration on the quasisyntomic sheaves $THH(-)^{hC_n}$ and $THH(-)^{tC_n}$ where $C_n$ is the finite cyclic group of order $n$. These give cohomology theories that interpolate between the quasisyntomic filtrations on $THH$ and those on $TC^{-}$ and $TP$, and as it turns out can be expressed in terms of sheaves already of central importance to the study of the cohomology theories introduced by Bhatt, Morrow, and Scholze.
As applications, I will explain how these filtrations can be used to extend the results of Angeltveit, Gerhardt, and Hesselholt on the K-theory of truncated polynomial algebras, and a result on topological restriction homology.