Number Theory Seminar/ Representations in Arithmetic: Kiran Kedlaya

As its name is meant to suggest, the subject of p-adic Hodge theory was historically concerned with the relationship between different cohomology theories attached to p-adic algebraic varieties. Within p-adic Hodge theory, the concept of a perfectoid space (discussed in my PIMS lectures) arose quite naturally and has led to improvements in the subject which were in some sense "expected".

However, it also had several "unexpected" applications rather far afield. We'll survey three of these: Deligne's weight-monodromy conjecture (Scholze); Galois representations associated to torsion cohomology of arithmetic groups (Scholze, Caraiani-Scholze); and the direct summand conjecture of commutative algebra (Andre, Bhatt).