Topological cyclic homology is a variant of cyclic homology, based on the idea of taking Hochschild homology relative to the sphere spectrum. It is defined for arbitrary noncommutative rings $R$, and there is a natural map $K(R)\to TC(R)$ from the algebraic K-theory of $R$. In some sense $TC(R)$ is the closest approximation to $K(R)$ by a cyclic homology-like invariant.

I will explain some recent results which show that the relationship between $K(R)$ and $TC(R)$ is extremely tight, when $R$ is any $p$-adically complete commutative ring and we take mod $p^n$ coefficients. Removing the commutative hypothesis from these results is an interesting and difficult problem for the future.

This is based on joint works with Bhargav Bhatt, Akhil Mathew, and Matthew Morrow.