Abstract

In this talk, I will discuss recent progress in motive theory and its connections to p-adic cohomology. I will begin by explaining the original motivations for developing motive theory and provide an overview of mixed motives. Following this, I will introduce Voevodsky’s triangulated category of motives, known as $DM$. I will then explore the broader category of motives with modulus, $MDM$, which extends $DM$, and explain how it relates to p-adic cohomology. Finally, I will propose a conjecture about the behavior of Hodge-Witt cohomology within the $MDM$ framework, and show how this conjecture leads to the existence of a good integral model for rigid cohomology. This approach builds on a modified version of the methods developed by Bhatt and Snowden.