I will discuss the theory of syntomic complexes (due to Bhatt--Morrow--Scholze and Bhatt--Lurie) and their identification for regular schemes (joint with Bhatt), inspired by some work in topological Hochschild and cyclic homology.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/10868