Bhargav Bhatt

Title: De Rham cohomology for algebraic varieties

Abstract: The topological cohomology of complex algebraic manifolds enjoys
an extremely rich algebraic structure, traditionally the subject of Hodge theory.
A key piece of this structure is Grothendieck's purely algebraic take on de Rham
cohomology (together with the isomorphism between topological and de Rham
cohomology for any manifold). I will begin by reviewing this picture, and then
explain how the algebraic perspective leads to a "de Rham" description of the
cohomology for algebraic varieties with arbitrary singularities. I will then discuss
some consequences for Hodge theory over a p-adic field.