P-adic Milnor K-theory, Syntomic Cohomology and Chow Groups

In this talk, I will explain how Milnor K-theory, a recent version of syntomic cohomology due to Bhatt, Morrow and Scholze and Chow groups can be related through the Bloch-Kato and Gersten conjecture. I will focus on the mixed characteristic situation. I will also explain how these concepts appear in the study of Chow groups in families and state a recent conjecture in the case of rationally connected varieties. 

Finally, I will propose how the mentioned new definition of syntomic cohomology may be used to overcome difficulties in defining a 'syntomic homology theory with duality'.