This paper incorporates the theory of Hochschild homology into our program on log motives.
We discuss a geometric definition of logarithmic Hochschild homology of animated pre-log …

These are notes for a course given at Princeton University in Fall 2022. The goal of this
course is to ultimately explain some work in progress of the author with Jacob Lurie on …

The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate
twists of Geisser–Sato–Schneider on regular p-torsion-free schemes. Our methods apply …

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are
representable in the category of logarithmic motives. As an application, we obtain Gysin …

In this article, we study the topological Hochschild homology of $\mathbf {E} _3 $-forms of
truncated Brown-Peterson spectra, taken relative to certain Thom spectra $ X (p^ n) …

We give a new formula for real topological cyclic homology that refines the fiber sequence
formula discovered by Nikolaus and Scholze for topological cyclic homology to one …

We refine several results of Bhatt-Morrow-Scholze on THH to THR. In particular, we compute
THR of perfectoid rings in which 2 is invertible. This will be useful for establishing motivic …

We study the algebraic $ K $-theory of rings of the form $ R [x]/x^ e $. We do this via trace
methods and filtrations on topological Hochschild homology and related theories by …

We show that de Rham--Witt forms are naturally isomorphic to $ p $-typical curves on $ p $-
adic Tate twists, which revisits a question of Artin--Mazur pursued in the earlier work of Bloch …

Algebraic geometry in mixed characteristic Page 1 Algebraic geometry in mixed characteristic
Bhargav Bhatt Abstract Fix a prime number p. We report on some recent developments in …