Abstract

Classical Hodge theory can be interpreted as the relationship between the topology of manifolds and the cohomology of differential forms. In arithmetic geometry, the parallel subject of p-adic Hodge theory relates Galois actions on etale cohomology to differential forms. Recently, this subject has been overturned by a series of developments culminating in the construction of a class of objects called “perfectoid spaces”. Without trying to give too many formal definitions, I will indicate some of the key ideas that go into the theory; as time permits, I’ll also mention some of the directions in which the work of Scholze carries this theory far beyond its point of origin.